Not allowed.
And just for trivia, they actually suggesting human shouldn't be totally naked in sex activity.
The reasoning for this because they're not a animal and there's angel that witnessing your daily act.
This is false. Although Islam emphasizes modesty, even in privacy, there is no requirement for a married couple to be fully clothed in each other's company at all times.
I'll provide answers from a Sunni standpoint:
Is it compulsory for women to wear a Niqab (Covers everything but eyes)?
Difference of opinion exists on this issue. Some scholars say yes, it is. Some say it is compulsory in a place where a woman's safety is feared. Others say it is not. The basis of this is from the hadeeth that describes how 'Aisha RA covered her face when a companion of the Prophet PBUH approached her when she was stranded alone in the desert.
Is it compulsory for women to wear a Hijab (Covers hair and neck)?
Yes. The Qur'an explicitly commands believing women to cover themselves. The minimum extent of coverage would only allow the face and the hands to be visible. Covering is only required in front of unrelated men, though.
Is it compulsory for women to wear a Burka (Covers everything, including eyes)?
No. But as with everything in the religion, circumstances shape obligations unless explicitly mentioned in the Qur'an and the hadeeth. Covering the eyes is not mentioned in the hadeeth, for example, but some scholars in certain areas recommend doing so. This has everything to do with the men in that particular society who cannot control themselves and nothing to do with the women themselves.
Do women have to pray in separate rooms as men?
No. In Makkah, there is no barrier separating the men and women either. The barrier came about during the caliphate of Umar RA because of circumstantial reasons. Men are obligated to pray at the mosque. Women are not. Also, children are usually in the care of women and it is considered disliked to forbidden (depending on opinion) to bring children younger than the age of 7 to the mosque. The reason a lot of mosques do have separate areas is because of not just convenience (as men are obligated to pray and women are not, entrance and space for men is more important than entrance and space for women) but also to prevent intermingling of the sexes.
Are women equal to men as legal witnesses?
This depends on the case. In some cases, yes. In others, no.
Are women allowed to work?
Yes. The wives of the Prophet PBUH ran their own businesses. The restrictions in some areas are only due to circumstance (or a misguided sense of 'protecting the women'), not something innate to the religion.
Are women allowed to drive?
Yes, they are. The opinion in Saudi Arabia is unique to the country. I suspect it is mostly due to the culture of the country rather than for any innately religious basis. Many women who wear the full face veil, for example, can be seen driving in many Muslim communities outside of Saudi Arabia.
Are women allowed to serve in the army?
Yes, but men are obligated to. Women are not obligated to.
Legal issues
Should thieves have their hands cut off?
Under Islamic shari'a, yes. The severity of the crime would warrant this but this is not the first penalty. So if you steal a loaf of bread, you don't lose your hands. The wronged party also always has the option to pardon or demand repayment in another form.
Should liars have their tongues cut off?
This also depends on the case. There is no explicit command to cut off a person's tongue but Islam follows Judaism in the concept of punishment, where an eye for an eye, a tooth for a tooth, etc. are allowed. In cases of slander, cutting off the tongue has been a proscribed punishment but this is also found in other Middle Eastern communities, such as the Assyrian Christians of Iraq.
Should muslims who have converted to Christianity be punished, and if yes - how?
Should muslims who have converted to Judaism be punished, and if yes - how?
Should muslims who have converted to any non-Abrahemic religion be punished, and if yes - how?
Should muslims who have turned atheist be punished, and if yes - how?
The answers to all this are the same. Apostasy in Islam, under an Islamic shari'a, is akin to treason. A person renouncing Islam is the same as someone renouncing the Constitution of the US. The punishment is death. In the Hanafi school of jurisprudence, the apostate woman can be imprisoned instead of being put to death.
The same punishment applies to political traitors, being those that still accept Islam but side with the enemies of Muslims to kill Muslims.
Should atheists be punished for being atheists, and if yes - how?
Should Christians be punished for being Christians, and if yes - how?
Should Jews be punished for being Jews, and if yes - how?
Should individuals subscribing to non-Abrahemic religions be punished for subscribing to non-Abrahemic religions, and if yes - how?
There is no punishment for being non-Muslim as opposed to apostatizing to a non-Islamic religion. Under an Islamic state, non-Muslims are not considered citizens as they do not adhere to the Islamic "constitution", which means the shari'a. Their only obligation to the Muslims is payment of the jizyah, a poll tax, provided they can afford it and provided they are males (women, children, elderly, and the poor are exempt from the jizyah). They can be exempted from this by serving in the Muslim army. On the other hand, Muslims cannot be exempt from the zakat tax.
Non-Muslims have their own legal systems, their own courts, their own prisons, etc. Non-Muslim communities flourished under Muslim rule as long as the shari'a was in place.
Should muslims who criticize part of the Quran be punished?
The Qur'an is the literal word of God. If a "Muslim" criticizes the Qur'an in a manner that negates part of it, then that "Muslim" must repent from this and reproclaim the testament of faith. Otherwise, he has apostatized. Criticism naturally entails that there is some doubt as to the veracity of the message.
Criticising ideas some have derived from the Qur'an, however, is highly praised, especially if done well. Scholarly debates in Islam are usually about just that (and Islam has a healthy history of intrareligious debate).
Should atheists who criticize part of the Quran be punished?
Should Christians who criticize part of the Quran be punished?
Should Jews who criticize part of the Quran be punished?
Should others who criticize part of the Quran be punished?
No. Non-Muslims under Muslim rule criticized the Qur'an but were not punished for it. Non-Muslims can even insult Allah without fear of reprisal. Muslims are forbidden from insulting the gods of other religions, though. The only thing that Muslims won't tolerate is insulting the Prophet PBUH. Why? Because when one insults God, God has the right to seek retribution Himself, whereas when one insults the Prophet PBUH, the Prophet PBUH does not have this ability.
Is secularism compatible with Islam?
To an extent, yes, which means that a person can be Muslim and live in a secular country without him trying to change it. But in the long term, no, it is not, because Islam is a way of life, which includes politics. If a country is majority Muslim, some Muslims must strive for the establishment of Islamic law as the law of the land.
Is freedom of press compatible with Islam?
Yes. The caliph and the shoora are not some sort of divinely appointed party.
Is religious freedom compatible with Islam?
Islam does not allow people to convert out of it. It also does not allow non-Muslims to preach their religion to those of other religions. That means that Christians cannot send missionaries to convert Muslims/Jews. As for non-Muslims proselytizing to other non-Muslims of a different religion, that would depend on the laws of the religion being targeted for preaching. Remember, each religion or ideology is allowed to practice its own laws under and Islamic government.
Is free speech compatible with Islam?
Carte blanche free speech is not compatible with Islam.
Theology
Should Haddiths be followed?
Yes. The ahadeeth (sing. hadeeth) are graded depending on their authenticity, though. Some ahadeeth are found to be authentic if there is a continuous chain with completely trustworthy narrators. Others are not as authentic if they contain someone with a poor memory or if someone was relatively aged by the time they narrated the hadeeth. Some are deemed inauthentic if any one person in a chain is known to be a liar. The science of hadeeth is extensive. Furthermore, some ahadeeth are ranked as "tawattur", which is the same status as the Qur'an in terms of chain of transmission. This means that there is not even a shadow of a doubt in their authenticity. If someone denies them, they might as well deny the Qur'an.
Is the Quran without flaws?
Yes. The Qur'an is the literal word of God, preserved and protected since revelation.
Can parts of the Quran be ignored?
No. Everything in the Qur'an has a context and cannot be taken in a vacuum. Those who do so are in essence rejecting ahadeeth. Those who reject ahadeeth overall are also considered expelled from Islam.
Can some Haddiths be ignored?
See above.
Sex and social issues
Is abortion permissible (please specify up to which week, month, trimester if yes)?
Yes, only if there is danger to the mother. Otherwise, no.
Is contraception permissible?
Yes. Almost every form of contraception is permissible.
Is sex before marriage permissible for men?
No.
Is sex before marriage permissible for women?
No.
Are gay relationships devoid of sex permissible?
Legally and technically, yes.
Are gay relationships inclusive of sex permissible?
No.
Are gays allowed to marry?
No. Marriage in Islam is not the same as marriage in the West. The husband has the obligation to maintain his wife. The wife has no such obligation. How would this work in a male/male relationship?
Are gay couples allowed to adopt children?
If by "gay couples", you mean men or women who cohabit without sex, then yes.
A note on homosexuality:
It isn't homosexuality that is prohibited in Islam, as some people have these urges and others do not. It is acting upon them. Sodomy is not allowed in Islam, whether between a male and female or a male and a male. Also, note that sodomy is not restricted to anal intercourse.
Is anal sex permissible for hetereosexual couples?
No.
Is oral sex permissible for hetereosexual couples?
No.
Is non-missionary style sex permissible for hetereosexual couples?
Yes.
Is masturbation for pleasure permissible in Islam?
No. The only exception would be if someone is unable to control himself and comes to the brink of committing extramarital or premarital sex.
There's a general rule in Islam: under extreme circumstances, many things that are forbidden become permissible. For example, pork becomes permissible if a person has no food and is starving.
Is in-vitro fertilization acceptable?
Yes.
Is selective abortion acceptable (to determine gender)?
See above. Abortion is not acceptable unless the mother's life is in danger.
And what do you mean by "to determine gender"?
Is selective abortion acceptable (for the purpose of removing genetic and other diseases/deformities)?
See above. Abortion is not acceptable unless the mother's life is in danger.
Is Human genetic engineering acceptable?
Still under debate. Personally, I believe eugenics to be morally reprehensible if used for determining aesthetic features, like skin/eye/hair colour. If it is used to prevent disease, such as genetically eliminating cystic fibrosis in utero, then there is no harm.
Misc.
Are interest rates un-Islamic?
Yes. Interest is worse than premarital/extramarital sex. Interest, in a ranking of sin, is worse than even incest. The reason is that extramarital affairs may destroy families, but interest destroys societies. The current financial crisis can easily be seen as an outcome of an interest based economy.
Are dogs allowed as outdoor pets?
Yes.
Are dogs allowed as indoor pets?
Depends on the school of thought. The Malikis allow this; the other schools do not.
Are pigs allowed as indoor pets
Are pigs allowed as outdoor pets?
Pigs are unclean animals. The less one has to deal with them, the better. We are not even allowed to wear clothing or items made of pig skin.
Is it permissible to consume pork?
No, extreme circumstances notwithstanding.
Is it permissible to consume alcohol?
No, extreme circumstances notwithstanding.
Is it permissible to cook your food in alcohol?
No.
Is it permissible for men to pluck their eyebrows?
No. Unibrows, where a bridge of hair forms between the two eyebrows are not considered part of the eyebrow and can be removed.
Is it permissible for women to pluck their eyebrows?
No. The only exception would be if not doing so would make the woman undesirable to her husband.
Is it permissible for women to shave their leg/arm/chest/back hair?
Yes.
Is it permissible for men to shave their leg/arm/chest/back hair?
Yes, but it is not recommended.
A closing note on the legal system of Islam
In the legal system of Islam for the most part, actions are divided into a broad range of categories:
1. Obligatory, undeniable (fardh) -> meaning that the action is obligatory and to deny the obligation is to deny the religion itself; fasting in the month of Ramadhan falls under this. Note that it isn't that not doing the action expels one from Islam; so if a person does not fast in Ramadhan without a valid excuse, that person is still a Muslim but a grave sinner. If the person denies the obligation of fasting altogether, then that person has rejected Islam.
2. Obligatory, deniable (wajib) -> meaning that the action is obligatory but its obligation can be denied, based on scholarly differences. For example, there is an extra prayer in the night prayer called the "Witr" prayer. In the Hanafi school of jurisprudence, this prayer is obligatory. If someone denies this obligation, then they are not considered expelled from the religion nor are they considered sinful if they have evidence. The other 3 schools of thought and the Salafis in general do not consider Witr to be obligatory, for example.
3. Recommended (sunnah) -> these can be further subdivided into muakkadah (strongly recommended) and ghair muakkadah (desired). These are derived strictly from prophetic actions as well as the actions of the companions of the Prophet PBUH. These do not include things that the Prophet PBUH commanded be done (these would be obligatory actions). An example would be to fast Mondays and Thursdays. The Prophet PBUH did this but did not command others to do so. Another example would be the late night prayer (tahajjud); the Prophet PBUH did this but it is not obligatory for anyone. Another category, mustahabb, which means desired or liked action, may not necessarily find direct evidence in the Qur'an or hadeeth or the sunnah at large, but may be derived based on the 'spirit' of the narrations. For example, it may be considered mustahabb to do things in threes unless otherwise specified.
4. Optional (nafl) -> these are actions that are supererogatory actions that are extensions of the sunnah actions. For example, to fast on a Wednesday would be considered nafl. Sometimes the recommended actions are seen as a subcategory of optional (nafl) actions.
5. Permissible (mubah) -> these are actions that are permissible, such as reading a book or writing or going out for dinner, etc. There is no inherent reward or sin attached to any of these actions. It is intent that can make any of these actions into rewarding or sinful. For example, going out to dinner with the family with the intention to bring them together and increase love for one another is rewarding. On the other hand, going out clubbing is sinful.
6. Disliked (makrooh) -> these actions can be further subdivided into tahreema (close to prohibition) and tanzeeha (not as bad). These are actions that are not necessarily sinful but may be if repeated. An example would be to continuously and willfully omit sunnah actions. Another example would be to not remove the arm pits and pubic hairs. The latter can even become prohibited and sinful if the person does not remove them for 40 days or if they get longer than a grain of rice (whichever comes first).
7. Forbidden (haraam) -> these include things like drinking alcohol, lying, murder, etc. Some come with punishments, some do not. For example, not praying 5 times a day is haraam, but there is no specified punishment for this. Also, for almost any crime where another party is wronged, the maximum penalty that is afforded to the wronged party is what is specified. For example, if a man murders his neighbour, the neighbour's family has the option to have the man executed but that is the most extreme permissible option available to the family. Families can also get blood money, seek imprisonment of the murderer, or even forgive the murderer. Interest, and usury in general, is also forbidden.
Not everything is black and white in Islam. You'll sometimes even get remarkably different answers depending on which school of thought one adheres to, the scholar the question is asked to, etc. but there are some set rules and those are the ones explicitly mentioned in the Qur'an.