"I am washing my dog" is fine, BUT it does sound a little like the dog is part of you, like your arm or leg. More common is to say "I am washing the dog". "I am bathing the dog" works too (the verb here is bathe, not bath, so "I bathe my dog"). If I was saying it, I would actually say "I am giving the dog a bath" or "I have to give the dog a bath".

If you put him into the bathtub full of water ( like you might do with a baby), you can use the same expression as in Portuguese  : "I give my dog a bath".

"Cachorro" is a "puppy" in English, if he's a very young dog.