1. What was the story with the young girl and Char?. Why did Char take her and get her to pilot the mobile suit?.
She got taken in by his whole "souls and gravity" speech, for one. She was also infatuated with Amuro, but he was already with Chan. Essentially, it's part rebound and part "I'll make him jealous by going with this other guy." That, and she's pretty much in it for herself only, so she doesn't have any real sort of ties with the Federation.
As for Char making her a pilot...well, she's a Newtype (and a powerful one at that) and he's trying to start ****. Might as well use everyone he can.
2. What was up with Gyunei being so hung up on the girl to the point of wanted to turn on Char?
It's part jealousy for her being a better pilot than him and part him having a thing for her. He essentially wanted to out-Char Char. You saw how well that worked out.
3. Was it ever explained how Char survived the AEUG/Titans final battle from Zeta Gundam?.
Nope. Presumably it was going to be explained when he showed up in ZZ, but when CCA got greenlit, Char got cut out of it.
4. When did the AEUG disband?.
After the end of ZZ. Most of the AEUG went on to form Londo Bell, so it's almost as if they never left.
5. When did Char actually become leader of Neo Zeon?. It feels like there was a massive gap in between the end of ZZ and this film, was any of this answered in novels or anything?.
It's actually a different group with the same name, though I don't doubt that there are a lot of members from the first Neo Zeon that are in the new one. As for the gap...I dunno, maybe it's to separate it from ZZ? I haven't read Beltorchika's Children, so I don't know if it says anything on the subject.
6. Char worked with Amuro in Zeta....everything was fine and now he wanted to kill Amuro?.
Because the whole point of CCA is, essentially, to finish up the whole Amuro/Char rivalry that started way back in the One Year War. Really, if you look back at everything you've watched thus far, it's all kinda leading up to this...though it's not always apparent. Keep that idea in mind when you get around to Unicorn.
...okay, so there's sort of an in-universe reason for it as well. Char has always (at least as of the post-OYW era) been pro-spacenoid. He worked with the AEUG because the Titans were oppressing spacenoids. Now when CCA comes around, he's still working to better things for spacenoids...it's just that his idea for helping is forcing everyone to become one.
7. What was Char talking about Lalah becoming a mother to him?. I thought it was his lover.
She was. I think it was more of a metaphor for him developing as a Newtype and as something other than a revenge-driven soldier. Then again, it could just simply be that mommy and/or daddy issues are A Thing in anime.
8. Why did some of the Neo Zeon pilots end up helping Amuro to veer the Axis asteroid away from earth?.
It's sort of a "well, we don't like how Earth is doing things, but we don't want to see it get its **** wrecked" deal. There may or may not have also been some influence from Amuro and his psychoframe, though that's more of a personal theory.
9. How exactly does that psychoframe light help move the asteroid as well?.
It's Newtype Magic and/or Anime Bullshit. Just go with it.
10. The ending didn’t really clarify if Char or Amuro died.
No. No it did not.
Will we ever learn the whereabouts of Kamille and Jewdough by the end of Victory Gundam?.
I haven't seen Victory so I can't say for sure, but I don't think so. As far as we know, Kamille's still on that beach (or not, thanks to the Zeta movies) and Judau's still truckin' his way towards Jupiter.