CAN WE REQUIRE LIABILITY INSURANCE FOR DOGS? CAN WE REQUIRE THAT WE BE NAMED AS CO-INSURED?

There is no reason why a landlord cannot impose such obligation on your tenants, but such obligation does not negate any community owner obligation to be well insured and to protect the community from harm. If you do impose such obligation, be sure such insurance is available and possibly recommend how a tenant may procure such insurance (to make case of compliance possible).