1. The Second Impact somehow altered something about the children being born right after it occurred, thus making them the best possible candidates for Eva pilots.
2. Unit-01 only accepts Shinji. Unit-02 only accepts Asuka. So even if they're mentally screwed kids, you can't do anything about that.
3. Second Impact occurred when humans tried to make a contact with Adam. Its awakening caused a massive explosion on Earth, and we know what happened afterwards. As for the Lance, it was accompanied together with the angels' arrival on the planet, so it was with Adam from the start (Lilith's was destroyed before it arrived).
4. First Impact was literally a collision between Earth and the Black Moon, which contains Lilith.
5. Yes, though technically it is just merging of humanity's souls into one rather than a literal impact of something that will kill humanity. Shinji and Asuka were the first to appear because of their strong will to live. Shinji was screwed up so yeah choking Asuka the first time she saw her wasn't really that surprising
6. You can say that Angels were aliens, or ancient life forms long before humans existed. Humans were Lilith's children after all.
7. Kaworu is basically the same with Rei in that he was created containing the seed of life, in this case he was fused with Adam's DNA. His sole purpose is to make contact with Lilith to start Third Impact, which SEELE wanted from the very beginning.
8. Yes. They believed that Adam should be the one who populated the Earth instead of Lilith, thus they believed humanity carries the original sin and needs to be purified by merging the souls into one with god.
9. He just wants to be merged with Yui. That's why he went along with everything.
10. Kanji was a double agent, so she became Nerv's, and Misato's, enemy. Though it is uncertain that Misato was the one who killed him.
11. There are already robots you know. They just aren't the cool badass ride-able form