The night the Doctor first met her, why in the world was a kid that age home alone? I only have one friend who watches Doctor Who and he watches in long marathons, months (even a year) apart. Try to talk with him about something specific and he's all "Huh?" I've wondered about this for a long time, on and off, and the computer was handy when it came to mind this time.
The crack erased Amy's parents from existence. Amy herself still existed because when a crack erases something, that something's effect on the universe, in this case Amy, remains (another example is the Byzantium remaining crashed after the Angels that brought it down were erased by that crack). Amelia was living with her aunt Sharon when we first saw her. Now as for where Aunt Sharon was, that was never explained. All we know is that she went somewhere that night and left Amelia alone.
Maybe the crack erased her parents just that night, so her aunt had no reason yet to be there. Would explain why the TARDIS landed at the same time.
Didn't she give the Apple to the Doctor as a child and he used it in a fashion to prove he had travelled in time?
No, she said it was something her mom would do, did it herself, and gave it to the Doctor, which is why it was still fresh when he gave it back to her as an adult.
Actually, it's quite striking how many questions are addressed by the show itself when one starts paying attention.
I promise, I actually do pay attention. It's just...I dunno. Maybe I figured it out before and it didn't stay with me? Some of what you're saying sounds familiar now...maybe (Talk about my friend and his memory...we're probably both in the same boat.)
You remember him asking Amy "you ever wonder what happened to your parents"? It's a fairly involved speech about the cracks and their erasing people from time.
Speaking of lack of attention - I never understood who the voice was in the TARDIS that then made it explode? Were we just meant to assume it was one of the silence or was it never clearly explained?