Probably. I know if you don't pick the right dialog options for when you meet Yuffie she steals your stuff. It's a good idea to save before you run into her.

I remember leveling up the three in my party at a time every two or three levels before each major boss fight. That's pretty much how I won. I can never remember who were in my party at a time or how consistent it was outside of Cloud, but you could probably get away with sticking with a certain group throughout most of it. I know Barrett will have to fight solo in one part of the game so it'll probably be a good idea to keep him strong.

I never fought Don Corneo's bird so I can't help there.