1. Is consensimency genetic? Is it passed down from parents?
No, it's not.
2. Are there any particular character traits that would common between consensimens?
Nope! It's a completely physical SF and does not come with a set of personality traits a character must have.
3. Does the ability affect the consensimens if they are touching an animal, say a dog, or a plant?
It's really a feature that has to do with people and not animals/plants. I can see it maybeeee extending to other magical creatures (house elves, centaurs, trolls), but not non-magical animals.
4. Does consensimency affect normal magical ability at all?
No. It's strictly touch-based. The only time I can see it possibly hindering magical ability is if a healer was a consensimens and treating a patient, or if a patient was a consensimens. As long as the magic isn't touch-based, it shouldn't affect them.