Blur,
Even in New York you can shoot a dog if it is threatening your person or your property. Now you'd have to clearly be able to prove that the animal was a threat, but yes. A dog that is on your property threatening you or livestock can be shot with "no liability on such persons in damages or otherwise for such killing" Just clarifying the law for you there. They wouldn't take your gun.
As for tex's sika, There's a good chance it was pretty wild. Some of those ranches are huge. Also, he said 50 miles is the nearest ranch, which means this one's been out quite a while.
-Jake