1. doubt it...Laura isn't really a leader type and hasn't shown any real skills in that regard. She will most likely either be the one who gets her hands dirty so the rest of the team doesn't have to...or we will be treated to yet another "you are not a weapon" arcs
2. She is the most likely candidate after Scott. But I could also see Kitty doing it if Jean doesn't step up.
3. I think that would be interesting, very interesting actually...but I don't think either Emma or Jean would be comfortable with the idea. Phoebe and Irma would be good secondary options (Celeste still hates her)
4. small training scenes...sure, no reason not to when both teams are at the base. Their adventures are almost certainly going to remain separate though
5. possible, but I don't think it will happen. The only reason that was interesting at all was because Scott was there for it to play off of. Plus I still think she was just trying to change her own future and Hank was the first available option.
6. would be awesome, and I still feel it makes more sense for Rachael to be with Scott instead of Logan, but nothing so far would indicate it happening anytime soon. Maybe next X-men cross over event.
Why should she go to his hated rival's school ?
For one her best friend Kitty's over there. She even agreed with and helped Kitty help Scott and Jean during BoTA.
What exactly is she even doing at the school? You say teaching but I've barely seen her do anything but argue with Storm as of late.