Hudson states in the first episode that K-9 officers are euthanized if their handler dies and they are not claimed by someone. This is not true. It costs thousands of dollars and years of work to properly train a police dog. Healthy dogs are re-paired with new handlers, and when they are retired, if their current handler does not choose to keep them, they are put up for adoption, the same as any other dog.
Rex is obviously not a police dog. Anytime he jumps at a suspect, he either just lands on them or bumps into them. A real police dog would latch on with his jaws the second he was able.