Hi B.,
I have a 5 year old son who does the same thing. Our two year old cat is incredibly mild-mannered, and has put up with a lot from him. She only nips when she's playing, and has only scratched him on three occasions, for which I count my blessings.
I haven't gotten a reason from him as to why he's so rough with her (and downright mean at times), I think he's just exploring his curiosity. However, we had to set some pretty strict rules.
First off, we'd only let him pet her when we were holding her. She's an attention junkie, so this was very easy to do. After he learned how/where she would let him pet her, we allowed them to be on the floor together. It took a few months, but he learned that she had to come to him. (If he went over to her and started playing, he got a time-out and she went to get some peace and quiet in my room). We also only allowed him to play with her (string toys, balls, etc) when we were right there with them.
This worked for a while, but then he started back into the rough housing again (and I mean, pulling her tail, shoving her around, etc). My son, like your daughter, is sweet and caring towards everyone he meets, and I was amazed that he could act like this. Finally, we told him Katie was going to have to go live with a different family. When he saw how sad this made his father (the cat was a gift to him from my son and I), his behavior turned around. We sat down with him and asked how he'd feel if someone started treating him this way. Once he understood that his actions were hurtful to the cat, and that if he didn't stop treating her this way she'd have to go live with someone else, he became a lot more careful with her.
He still plays with her, (which he should, she's part of our family), but he's much better about it. He'll say "Look Mom, Katie came to me!" and then he knows it's ok to play with her. It took us alot of patience and repetition (and a few bandaids when Katie wanted to help it sink in) but it worked.
As far as other pet owners, I like the previous suggestion of having them tell your daughter their rules; everyone is different, and everyone raises their pets a different way. Maybe have her ask upon arriving at their house what is ok. It gives her control of the situation, and may make her understand the need for being gentle a bit better than just hearing "don't do that".
Sorry for such a long-winded answer. Patience through these things is tough, but I know you two will get through it. :)