And when I rename my pet, nothing about it will change...stats and such. And will it still be on high score lists with a different name? Thanks in advance

Correct, nothing about your pet will change but the name.

There are threads here with lists of available real names and real words: lock plz! Available Real Word Pet Names 3.1

or you can search a name here to see if a pet exists already with that name and then check if it might be on an inactive account from there

thank you very much

Also if I was to swap 2 of my own names nothing would change with the pet still

Right. Just the names would change, so Pet1 would have all of its stats and TC items and all that and just be named Pet2 now instead, and vice versa