Some verses from the Dhammapada;

> 129. All tremble at violence; all fear death. Putting oneself in the place of another, one should not kill nor cause another to kill.
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> 130. All tremble at violence; life is dear to all. Putting oneself in the place of another, one should not kill nor cause another to kill.

Though a dog might fear pain, is it possible they'd fear death (or a person's violence resulting in their death) even more?

Buddhism doesn't recommend "mercy killing". It's considered very wrong (unforgivable) for a monk to even recommend the alleged "advantages" of suicide, to someone who's in pain.