D.,
I have had the nightmare neighbor kid and know how it is - coming into our house uninvited and eating out of the fridge, beating up our toddler, trying to sneak off with my daughter and the doctor kit, peeing in our yard in front of our children. I get how frustrating it is. But I took the situation as a character building opportunity for myself and my children.
Yeah, the neighbor girl isn't very nice, but everything you mentioned is rather petty. If it bothers you that much, talk to the parents about it and clear the air. Try to see if you can turn over a new leaf with the neighbor girl - demonizing her won't improve the situation. If things don't improve, you can explain to the neighbor girl that she needs to be nice to everyone in order to play at your home and be consistent in sending her home if she is hitting, kicking, name calling, etc.
As far as your daughter goes, this is an excellent opportunity to talk about how everyone makes choices and we don't know why some people make lots of bad choices. It's a great opportunity to talk about forgiveness, setting a good example, and how other people's poor behavior has nothing to do with our self worth. You can only protect her from "bullies" to a certain degree - and should if they are getting physically violent or cruel. But your daughter won't be four forever. She will endure the nightmare of middle school and high school. She will meet mean people in the workforce, as roommates, and possibly as dates. You can't be there to shelter her from every look, every word, every passive aggressive gesture. All you can do is arm her to deal with it, which is a much greater and long lasting gift.
Good luck,
S.