Is it sinful to curse an animal, whether in jest or with the intent to insult, and does this fall under Allah's or the Prophet's prohibition of animal abuse?

The prohibition against torturing animals is a prohibition against using anything with a soul as a target. As for mockery, the prohibition against it was revealed concerning humans in His (the Most High) saying: "O you who have believed, let not a people ridicule [another] people." Therefore, insulting an animal or mocking it is not considered forbidden, but a Muslim should utilize their time in what benefits them and accustom their tongue to speaking good.