All pets need to be on a leash when they are outside under the control of the homeowner. If your pet is leashed and if they bite someone or damage property, it's the homeowner's liability and not the HOA's.
Dogs: City of Omaha Sec. 6-147. - Restraint—Dog. It shall be unlawful for the owner of any dog within the city to fail to keep his dog securely restrained or otherwise confined in or upon his premises in an enclosure sufficient to contain the dog. Homeowners need to follow this guidance.
If you walk your pet, pick up "pet waste" that your pet leaves behind.