One thing I have noticed is that the traits for pets are clearly separated into "good" and "bad" traits.
No such definition exists in Sims: if you have a rough pregnancy, fail to train a toddler, or do poorly in school, you have just as much chance of getting a desirable trait as you do of getting one that is less desirable. I have repeatedly had sims receive the Genius trait after doing poorly in school, for example. The game just doesn't make a distinction between traits as far as "good" or "bad."
But, from my limited experience thus far, it seems that baby pets are far more likely to get undesirable traits to start with. They then have to unlearn these and learn better traits. Keep in mind that my experience is limited to adoption from neighbors: I've only just started to experiment with horse breeding, and have yet to get any other pet pregnant. So I don't really know how the parent's traits influence the baby's traits (if they do at all).
Thus, I'll soon be back with more info.