Sounds like an allergy to fleas. Funny to think that a dog which is synomomous with fleas should be allergic to them, but many dogs are. One of my dogs is the same. Basically use a very gentle wash, I use an oatmeal one to clean their coats, but don't scrub their skin as you can set up a dermatitis cycle of non-healing skin. Clean all their bedding and spray for fleas. Use a good quality flea protectorant on your beagle. I use Frontline liquid on their backs once a month. A watch them. They may also be getting bitten in the garden as fleas can live in dry sandy soil as well. If at any time you have any doubts about the cause or the healing take your dog to the vet for a comprehensive checkout.

The little black spots sound like flea poop! Get puppy dipped but have the vet do it if there is open sores!

Frontliine, Advantage and other toxic chemicals POISON our companion animals as they are absorbed through the skin. They also expose US and any children we have to toxic poison. Neems oil and Neems soap (both available at many Health Food type store....such as Fresh Foods, (and many will also order them FOR YOU, or you can run a Google search and order them directly yourself on line too) IS NOT TOXIC, OR POISON and WILL keep fleas, ticks and skeeters away from you and your animals. (Dogs, Cats, Horses) It can be safely used by humans as well. You can make a spray by mixing drops of the oil in water in a spray bottle (about 8 drops per cup and it will also keep flies off of your horse) It does sound like fleas to me, without being able to view the dog. You WOULD NEED to also wash the bedding in the Neems soap, and spray everywhere the dog has been in the house. The yard too should be treated, and I hate to suggest using a toxic substance, but I'm not sure how much Neems you would need to free that space from the dang buggers. You might try doing a Google on Non-Toxic flea and insect control. I probably would, assuredly if you know you do not even have fleas (!) have your vet give him a look over. It is possible that it could be an allergy to something other than fleas. Unless your bank account is excessively healthy, I don't generally suggest that you run allergy tests at the Vet. MOST vets will inform you that they can test but will also quickly share that it is expensive AND just like human allergy tests...they are NOT always conclusive! Better to begin reading the ingredients lists of your dog food, and consider that as your first possible problem. (if fleas are assuredly NOT the issue here) The vet can do a simple skin test, scraping to be sure there are not mites causing this, which it also could be...and the Neems would help with that too. The use of a high quality Oatmeal soap or raw oatmeal treatment (you do get it wet to apply) WILL very likely sooth his skin quite a good bit. I would strongly discourage you from using any sort of TOXIC substance, treatment, soap, dips, powders or collars on him, these products can also cause negative reactions to an already existing illness.