The book never states what Jill has been bullied about! While I don't think there needs to be a backstory involved, well, there still could be a reason. The book indicates that "what ten of the biggest boys and girls liked doing was bullying others!" Well, later in The Last Battle, Eustace describes Jill as "small". It may be possible that Jill is being bullied because she is small, in comparison to the ten of those big boys and girls.

I'm not really sure any of these are "unanswered questions" posed by the books... most of them seem to be trivial background details that aren't explicitly stated in the text because they aren't relevant to the story.

For example, if you really wanted to give a thorough explanation as to why Jill was being bullied in Act 1 (bearing in mind that bullying very rarely has a reason or purpose) you would really then have to pay off that explanation as part of Jill's character arc in Act 3, which hardly seems like it would be worth the effort.

When it comes to "unanswered questions", I wouldn't mind if what CS Lewis only hinted at in the books were fleshed out in a some way.

I was actually listening to the BBC Radio 4 Drama of The Last Battle, and in the prologue, Rishda makes an petition to Tash in Tashbaan (I'm assuming that's where, though I don't know for sure). Then he makes a monologue about his devised plan to "bring about the end of Narnia." It actually fleshed out his disbelief in Tash or any of the supernatural. The monologue is actually quite good for a radio drama, though I'm not sure how well it would work for a movie. Here's what he says in the monologue-          
"If I bring about the end of Narnia, it will be my own strength and cunning, not by any help of a gruesome vulture-headed god. But I will let Tash take the credit, since the Tisroc will need someone to rule the country for him. And who'd better than the servant of the devout believer in Tash?"

Some may have raised the question, "Why would Rishda, one of the most powerful men of Calormen, being turning away from Tash?" While one's beliefs is relevant to the plot of The Last Battle, I don't know if we actually need to know what caused Rishda to turn away from Tash or any of the supernatural.