There are ways that your pets can gain more traits and stop doing the things that annoy you: Reinforcement. There are two types of Reinforcement: Scold and Praise. Scolding a pet for something you don't want them to do will cause the likelihood of the Trait to decrease. Praising them does the opposite. Enough scolding and praising can lead to your pet 'learning' a new Trait, which will appear as a trained Trait. You can see the likelihood of the Trait going up or down after the scolding or praising. Don't Praise them for doing something bad and don't Scold them for doing something good otherwise it won't work.

For example: in the pictures below I have Sophie from the Fennel-D'Avoure family. I didn't make Sophie with the Non-Destructive Trait but when she wanted to chew on everything I gave her squeaky toys. After chewing on these I was able to either Praise or Scold her for chewing on the toy and being Non-Destructive. I praised her for it of course and after a few days Sophie learned the Trait and stopped wanting to chew on everything! Sophie was also praised for peeing outside and gained the 'Neat' Trait. I scolded her for peeing inside, Being a Piggy, and it decreased her chances of learning it. She doesn't pee inside anymore.
I'm currently training my other Fennel-D'Avoure pet, Niji, in Loyalty. You can increase this by having the pet Bring In a newspaper and then praising them for it.

She was becoming desperate to fill her Destruction bar.

So she took to the chew toy.


And then Anna praised her for her decision.


And so her Non-Destructive side increased.


Happy Simming!