If you are playing in any other world but Brindleton Bay - I believe they will only get sick if you put one of the pee/woohoo bushes on the sim's property (I don't think it has to be the one that came with Cats and Dogs). Without the bush - there are no other animals or anything that would make them sick. They fake attack birds and squirrels in the bushes and then get sick sometimes from that - it is usually the same sickness over and over from a squirrel bite - rabies or something - I forget the name of the sickness. They will leave a bunch of those feathers too all around the bushes.
I really dislike that pets in other worlds are so different from pets in Brindleton Bay - the sicknesses and strays are way overdone in Brindleton Bay (which is why I don't really like my sims to live there even though I love the town itself) but do not exist at all in the other worlds.
As for pets always being sick when you visit other sims - yes, that always happens- why, I do not know. If I am playing around a lot of pets - I always keep wellness snacks in the sim's inventory and give them to the sick strays or the ones my sim visits and it cures them.
