UNMARKED SPOILER QUESTIONS AND ANSWERS BELOW!
Doing this from my phone. So future apologies if the grammar or spelling is off.
UNMARKED SPOILER AHEAD!!! YOU HAVE BEEN WARNED!!!
1) Threelights explained this pretty well. However, there is the added fact that Machina, Ren and Izana are all from a town that bordered Rubrum and Milites. So it would frequently be fought over and be a part of each nation for periods of time. Machina accepted the White Tiger Crystal because he wanted the L'Cie power. And as it turns out the l'Cie who gave him the power focus was to give Machina the power to become a l'Cie.
Why would Machina accept the power of an enemy l'Cie and WORK for the enemy when it was common knowledge to him and everyone that Rem was on Class Zero's side? He effectively became her enemy when he did this... and like the dunce that he was, neither revealed his identity to anyone (for... dramatic effect?) and ultimately ended up making Rem do her best Aeris-meets-Masamune impression later on. How could he not see that coming if he knew she'd continue fighting for Rubrum?
2) First off, they note in Chapter 4 that Aria looks a lot like the Generals sister who passed away. So when he saw her shot he wanted to save her because she reminded him of his sister. She was grateful that he saved her so she wanted to help him.
That makes only the barest of sense. She still betrayed her entire nation, that always supported her, after ONE single guy from Milites proved he wasn't a total jerk... and that was ONLY because she was lucky enough to resemble someone he lost, which is such an insanely selfish reason. And she still took a bullet from Milites (honestly, I thought she was full-on dead until it was stated otherwise... that certainly looked like she was dead upon being shot, but that's just poor execution). It would be like someone joining the Nazis after being shot by them because one of them wasn't as bad as the others. They're still going after her friends. Heck, the General goes into battle to kill her friends and she's cheering him on the entire time. Makes no sense to me.
3) When l'Cie finish their focus or disappear the Crystals always look for a new l'Cie immediately. In the moment you were given the choice to accept or decline the Vermilion Bird Crystal Primus l'Cie disappeared so the Crystal was looking for someone to take his place. If Class Zero declines then it asks Rem, who accepts.
That logic makes so little sense. Do the Crystals ALWAYS need someone else to do their bidding? What if there was a time of peace? What if they couldn't think of anything for them to do? Why can't they give them a more all-encompassing focus? For that matter, why was Rem even a good choice in the first place, outside of bizarrely-specific plot reasons for maximum dramatic and tragic effect? They could've picked anyone.
4) Izana was kind of the spark that set off Machina on his path. His desire to protect Rem stems from the fact that he doesn't want to forget her like he did with his brother.
I never really understood that either. I never understood the entire "we forget the dead" stuff. Is that for everyone who dies, even those of natural causes? Why would making them forget the dead be a good thing when many times the cry of battle to victory is to REMEMBER the fallen (Remember the Alamo! Remember the Maine!) and to avenge them because the loss is personal? Regardless, it's revealed the puppet masters at the end can break these rules anyway (Rem and Machina don't forget Class Zero), so why aren't there more exceptions to this arbitrary rule that only exists exclusively to justify more arbitrary Ex machinas? (heh! MACHINA! I GET IT NOW!)
5) Aracia is actually a goddess called Diva. Her purpose is to keep sending souls to the gate of Etro so they will open up. That's why she keeps reseting the world and enacting Tempus Finis.
So she's a god with the power to alter time itself. And she's repeated these events nearly a million times or more. Isn't that the definition of insanity? Trying the same thing over and over and over and expecting different results? But, also, if she exists OUTSIDE of time, and is aware of the events that have happened, why does she not take more drastic measures to ensure more drastic outcomes? Why not just tell the students "if you do this, you will all die and the world will end" and then ensure they take a different path? She's a terrible goddess. If things get really screwed up, she can always just reset time again.
6) It's a hint of the future of the series. Has nothing to do with Type-0 but rather the series if it continues. The movie in Japanese is called "Type-Next".
Since Ace is involved, it either involves reincarnation or is just an alternate universe. Maybe. I don't know. It could honestly lead to nothing... like decapitating Tidus with an exploding Blitzball or something.
7) Because Rem and Machina were l'Cin they were granted eternal life. When Aracia learned that the world was saved by her children, she undid the l'Cie curse on them so they can live on but she couldn't save Class Zero. So instead she undid the forget the dead curse so they can remember all of them so they could tell their story to the world.
Sounds like something she could have done for Machina upfront so he could remember Izana... but didn't for some bizarre reason. So much of the plot hinges on this "forget the dead" plot, and they talk about the "benefits", but we never really see the benefits in action. In fact, it constantly has the opposite effect and everyone spends a great deal of time thinking about things they can't remember and wasting a lot of time on people they no longer care about because they find it so odd they don't care about them yet know that they should. Who even created that curse in the first place, and why did they think it was such a good idea?
8) That is Tempus Finis. The end of the world. The world essentially tries to end itself so it sends out those creatures to end it.
What causes Tempus Finis? It just comes out of nowhere. There is no instigating factor. Are all these powerful beings sitting there at the door going "okay, guys, we have to wait for the right time where the world is united and everyone is feeling happy and then we'll strike! Oh boy, can you imagine the looks on their faces?" It's yet another completely arbitrary event that happens for arbitrary reasons that don't fully make sense yet exists solely to raise the stakes again for no reason whatsoever. They could have made Tempus Finis something that happens for any number of reasons and it would have made as much sense. Oh, they're using nuclear weapons? Time for Tempus Finis. Oh, a major political leader is dead? Tempus Finis time. Oh, Eight choked on a hot dog? Better bring out that Tempus Finis.
9) They were all orphans of war. Aracia took them in and raised them. They get their magic from her and not from the Crystal. Hence why they could use magic at the beginning of the game.
That does beg the question of why so many don't question how Class Zero is the only class in the ENTIRE WORLD not affected by magic nullifiers. You'd think not just Rubrum but EVERY nation would be after these kids to understand their immunity to the greatest weapon of war known to exist, and yet it's given the same consideration as what uniforms they should wear to class and whether anyone else is concerned Rem's been coughing non-stop for, like, months now...
If you have more questions or want me to expand on my answers let me know and I would be happy to do it. Though I am on my phone so it will take some time to formulate a response until I get home. But I will do my best.
You're doing the Fal'Cie's work. Heaven help you.