Can you give your spots to someone else?

I don't think that's possible, seeing as no one really owns the spot. Not even the one who created it, and that person doesn't have a higher "status" than other fans of the spot either. There isn't really admins on fanpop spots, and everyone can contribute links, videos, images and anyone can suggest images for the spots banner and icon etc. There's really no limitation to the fans contributions to the spot, even if they didn't make it. So I highly doubt that there's a way for your sister to make you admin of the spot. The only thing you (and of course all other fans as well) can do to prevent the spot going out of control is report spammers or abusive users, and if you find dead links, duplicates etc, tell the user who posted it. Ask that user to change link and so on.

No. Not really. This is unfortunately not possible.

I would suggest talking to one of the F4 or when you want to change the main icon/banner use your sister's account ie get the password of her