Note: I didn't read anyone else's comments so excuse me if what I say is a repeat. First off I want to say I totally get you and as a matter of fact I'm in the same situation. My bf and I are going on 2 years and he just started bringing his kids around me in the summer because we weren't ready and we had no toys or anything in the house for them. Well my bf took his old bedroom set that him and his ex shared and put it in my spare room, bought toys, and said "it's about time". And honestly I wasn't mentally ready before being that im BP 2 and wasn't stable. Well at first I was shy as hell. One boy is 6 and one is 9 and really, children can be rude but I've learned to take it with a grain of salt. Kids will be kids. They've criticized my cooking to the way my house "smells". (I burn insense sometime). Now the kids are getting pretty close to me and for Christmas the 9 year old got me a bracelet that says "mom". I'm almost 30 and never saw myself having kids. My bf and I have decided to just see what happens in that department. I have some issues after stopping the pill a year ago though. Whole different story.
Now as far as your situation goes, I think the girl might be a little jealous that her daddy is showing another woman attention. When I was a kid I would get jealous when my dad dated. But you seem pretty intent on not wanting kids. You need to think about the future and what you want before you go further with this guy. He may be everything you've ever dreamed of but you have to know that the dynamics of your relationship are going to change big time. Having the kids around will change things. It's no longer all about you anymore. He's going to put his children at high priority and the question is weather or not you can share him with these kids.
I wish you the best of luck and really take the time to think about it!