Having children of my own I can easily understand why the 9 year old is their, probably the baby-sitter fell through. I'd probably set her up in another room with a movie, books, whatever she is interested in, unless she really wanted to play or hang-out with her parents. If she wanted to hang-out I might try to explain to her what is going on and let her chime in with suggestions for the players. She might really get into the problem solving aspects of the game and from my experience, people that coem to the game for the first time often have alot of insightful ideas un-fettered by preconceptions.
I wouldn't let her play her own character as a 9 year old may get upset if things don't go well for them.
I'd most likely tone down some aspects of my game, but the adult themes as they exist (e.g. evils creatures are very evil, torture, kill children etc.) or even need to be touched on can be described in an oblique fashion that adults can get but shouldn't freak out a 9 year old. Probably nothing worse than she's already been exposed to via the news.
If the 9 year old is rude or disruptive, well this is a matter of manners, which I'm sure her parents can handle.
[EDIT] Forgot the part about an adult. Not so interested in an adult showing up as long as they are interested in the game. Well aware how many feel about RPGs and don't need a wet blanket and won't tolerate snide comments. You come into my home you show respect for my guests and our hobby or you leave.