Well, my views on these:
1. In X2, we had just Jean. In X3, we had to contend with Jean and Phoenix. From the subbasement scene between Logan and Jean, I was under the impression that we were seeing Phoenix - a completely different character. As Xavier says, a being of desire. This isn't the same, timid, faithful Jean Grey we've seen before. This is something else. Therefore, different motives. Not too hard to grasp.
2. How did you get the impression that the X-Men were fairly new in X1? I thought quite the opposite. Actually, when Scott disapproves of Logan joining their team, for me it establishes that Scott knows he's the man in charge - which implies they've been through this before. He doesn't want Logan to upset routine.
3. Fair enough - in X3, Logan did heal faster. But maybe the Phoenix wasn't concentrating too much - as others have said, maybe she was taunting him? She knew she could destroy him in an instant - for such a beastly being, I would guess she likes to see as much pain and suffering as she can. To satisfy herself, she would have to make Logan work harder, and therefore to stay alive he would have to push himself to the max.
4. I would never classify Bobby and Pyro as friends - in X2, we could see they each had different views. (Food courtroom scene; Bobby - "I think you're the only one having a good time."; Under the Institute attack; Pyro - "He can handle himself, let's go!" X-Jet scene; Pyro - "You always do what you're told?") They obviously aren't the best of friends - and therefore it can be safe to say that there is already conflict between them. This divide is only strengthened when Bobby realises Pyro has betrayed them, gone back on everything the Institute stands for. Why would Bobby have time for someone like that?
5. You've skipped an entire film here. X2 is the gap between them - to link the two. Storm goes from, as you say "shy follower" - to empowered X-Men (in X2, taking charge of the jet, rescuing the kids and Xavier) - to the newly-found leader in X3. This change in roles - it's forced upon her. Xavier gone, Scott gone, Jean- who else is going to take charge? Storm, in my eyes, is the only one certified; she's been here the longest. And she knows it too. What's wrong with that?
6. I wouldn't say pigheaded; sure Xavier comes across as darker, but his pupil just killed another of his! What do you expect? Throughout X1 and X2, we see Jean is a "favourite" of his; he goes to the Senate with her, when Magneto comes to the train station, it is Jean and Xavier who go - and at the end of X2, Xavier talks fondly of Jean, more so than he has of any other. In X3, this is continued. Jean holds a special place in his heart - and now she's come back as something completely different! No longer the Jean he knows and loves so much. Of course, his motives will be different; his actions possibly questionable, but if you think about it, it's understandable.