I've completed Friends of the Animals once or twice, so I'll have a go at your questions, too.
- How many different misbehaviors are there for cats? And for dogs? What are they? The guide says there's about a dozen and mentions some of them but it doesn't seem to have full list nor a guide for this aspiration. (At least, not that I can find).
Here's a few more not listed by Brian_Z.
*shredding furniture (cats) - buy a cheap sofa or armchair, do not buy a scratching pole
*Jumping on counter (cats)
*Begging for food
I never looked for a list, though. Just kept an eye on my pets, clicking on them whenever they seemed to be doing something undesirable.
When you see the "Scold for..." social, go for it. I had the best luck with a raccoon. He was evil!
- Can you only train out misbehaviors with your own pets? Yes, only your own pets.
- Any other tips for this that I might not have thought to ask about?
My best tip for you, since you mentioned never having played Cats & Dogs before, is to create a separate file and play around with Cats & Dogs for yourself.
Part of the fun of playing a dynasty challenge is exploring new-to-us aspirations ot game mechanics and figuring out how to complete/implement them.
I first had a sim go on a Mt. Komorebi Excursion in a play-testing file, because I was afraid my sim might die rock-climbing due to my inexperience, haha.
Good luck.