Dogging it - N. American term - giving less than a full effort to a task.
Dog does not carry a gun. He does carry a large economy sized can of pepper spray. Bail jumpers must be returned to custody ALIVE or the bond is forfeited.
I believe the term "Dog" refers to his tenacity in tracking down the miscreants. His given name is Duane
The police are certainly authourized to arrest bail jumpers, but it's not their top priority. The bail bondsman is usually given 30 days or so after the bailee fails to appear in court to produce them or pay the cash. Its a lot cheaper to pay Dog $5,000 to track them down than to pay the court the $50,000 bail.
As stated earlier, he of all people should have known he had to go to court. It was at this point that I stopped watching the show.