I am a firm believer in “death with Dignity” for animals and people. So what your friend did was totally acceptable to me. No one should allow an animal in a terminal state to needlessly suffer.
I can understand the legal concerns with discharging a firearm and agree that its not a good idea in an urban setting, but in the country where there is little chance of an errant shot shot injuring the neighbors I think its ok and that should be legal also.
In Florida we are a gun friendly state (I own more firearms then some small countries) so perhaps we take a broader view on this, but I think the deputy over reacted to the situation and your friend should NOT have been arrested.
In Florida I have seen state troopers and deputies end the suffering of animals who were hit by cars and were obviously not going to survive. This was roadside in suburban areas as cars drove by, so I doubt your friend would have been harrassed like that here in the Sunshine state.