Does rabbit fur make namaz void? Are we allowed to have rabbits in islam?

There is no problem in it, one can have rabbit as a pet in his house.
Note: The body and the salive of the cat, rabbit, and hamster is ritually pure (Tahir), yet its urine, feaces, and blood are Najis

Also it will not invalidate your salat if it’s furs are not to much on your clothes.