(1) A county may collect a license tax in an amount which shall be determined by the appropriate governing body from the owners and harborers of dogs and may enforce such tax by appropriate penalties. A county may impound any dog if the owner or harborer shall refuse or neglect to pay such license tax. Any licensing provision shall comply with subsection (2) of section 54-603 for service animals.
(2) A county may regulate or prohibit the running at large of dogs, adopt regulations to guard against injuries or annoyances therefrom, and authorize the destruction, adoption, or other disposition of such dogs when running at large contrary to the provisions of this subsection or any regulations adopted in accordance with this subsection. A county adopting regulations in accordance with this subsection shall provide for an appeal process with respect to such regulations.