Do non-talking characters’ names get put in ALL CAPS when introduced?

Yes.

Any essential element should be put in all caps when it is first introduced, especially a character, regardless of if it is human or if it speaks. So essential animals’ names go in all caps, too.

Department heads look for capitalized elements to guide them and tell them where and for what their department is needed on a particular shoot. If you if you fail to capitalize something essential, you run the risk of the responsible department overlooking it, and it not making it into the movie.