"To shower the dog" is an unusual expression in English, even if it is grammatically correct. "She is going to give the dog a shower" sounds more natural.
It's sort of weird in french, too. I used "Elle va donner une douche au chien ce matin" just as you'd say "donner un bain a." And I can prove it's a real usage:
I'll use your expression until someone says for sure that" Elle va doucher le chien " is just as good or is not too wierd. Could a moderator reply please.
Why not the reflexive se doucher?
Because with the reflexive, the subject is the recipient of the action: Elle va se doucher ce matin. In this sentence, the dog is the recipient of the action.