Outside of dogs that keep digging holes in the yard, I really can't think of any. The occasional dog or cat fight in the yard doesn't really affect anything except sims stopping what they're doing to watch. You have to make sure you lock doors to control where they go - for some reason, NPC pets tend to ignore locked doors sometimes. Pets don't go to Uni lots.
The funniest animal though didn't come with Pets. It's the penguin from Seasons but he/she acts like a NPC and you can't get one to live with you.
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