I am interested in renting a home. The landlord advised I'd have to give 1 month's security equal to rent and a nonrefundable $500 deposit for my dog. Is this allowed? If my dog does not damage, why shouldn't I get my $500 back? Has anyone heard of this?

Welcome to the world of renting with pets. Essentially it boils down to landlords can charge what they want for rent. Your landlord is telling you that, in order to make renting to you desirable for him, he wants $500 extra. It's not a pet deposit, it's a pet fee. He could also charge a pet deposit on top of that, which would be refundable. And some landlords charge "pet rent", which is an extra X a month per pet that you have. Which may be a good idea if you are staying somewhere for a short time, but quickly adds up if you rent for long times.

But the other option is that the landlord can just say "no pets", so the question becomes "how much are you willing to pay to have your dog with you?"