I've understood (and of course, I could be wrong in my understanding) that Lev. 11 and Deut. 14 specifically prohibit human contact ("You shall not touch") and consumption ("You shall not eat") of the carcasses (נבלה) of the unclean animals/ fish/ fowl/ insects/ etc. listed.
But, I don't recall these scriptures ever stating that you could not touch one of those animals/ fish/ fowl/ insects/ etc. while it was still alive (and thus not considered a נבלה).
So, is it permissible to touch an unclean animal that is living (e.g., having a pet pig, snake, etc.)?