First off, I know you don't want the dog. But, now that the dog is here, please try to show some compassion for the poor critter. Put yourself in his place--living in a garage with 16 other animals, probably eating nasty food or little food and having your whole body itch and/or hurt..... Imagine what that would be like..... Hopefully that will put the smell into perspective--compared to that, a doggy smell is really not a big deal.... :)
Putting him on a good quality dog food will help. A good quality food will have at bare minimum, meat as the first ingredient should be meat--and preferably the first 2 ingredients should be meat, and you should be able to read the ingredients (like real people food), and shouldn't have anything labeled as a "by-product"; avoid dog foods that have a lot of fillers or grains (like corn, beet pulp, etc).
You may also want to put the dog on fish oil (1-2 caps per day), since this can help with itching and the omega fatty acids are good for him.
With our dogs, who are healthy & do have healthy coats/skin, if I can't bathe them (I use a mild, scent free puppy shampoo). If I don't have time to bathe them, I use about 1 to 2 T of baking soda, and rub it through their fur, then brush it out. The soda absorbs some of the oils from the hair (the oils hold the smell) and so they are fresher smelling after you brush the soda off the dog.
Definitely continue to wash the bedding, too. And if he lays in a particular spot on the carpet, you can sprinkle some baking soda on that spot, let it sit for a few minutes, and vacuum it up. I have an old spice jar with a "sprinkler" opening that I use for this purpose, and I added some dried lavender blossoms to it. So when I vacuum, the lavender scent is released, in addition to the doggy odors being absorbed.