Yes the cats will stress your chameleon out. The cat could possibly see the little guy as food and when you aren't around try and attack it. Try your hardest to keep the cat away from the cage.
You can keep one room cat free, it's not difficult. From what I've seen the cat would most likely not notice the cham, but the cham would keep a close eye on the cat. The cham will always be stressed if it can see the cat. Imagine if you were in the same room as a tiger.......would you be stressed?
my guy camo is totally cool with my dog but scared of others, its weird it like he is used to my dog as no threat, but i dont know if cats would be the same, i would just keep the cats away to be safe...
My cat has no interest in my chameleon. Nor does the chameleon seem to notice or even care he is around.
Though my cat is only inside to sleep and drink. Could be different with each animal, just watch carefully and perhaps keep the chameleon in your room with the door shut.