At his age, allergies would indeed be *very* rare, but they can actually happen.
To me, those pictures look like he may have pyoderma (staph intermedius) - a skin infection...did the vet do a skin scraping of these reddish areas and look at it under scope, bc that would be all it needed to diagnose pyoderma.
I can't imagine a thyroid problem in a dog this young, but has his thyroid been checked?
You should read this thread about allergies and how to approach them:
http://www.yorkietalk.com/forums/die...tart-here.html
Here are your options:
1) keep going to your regular vet multiple times, probably about 5 more times at least...with no solution. This would probably amount to around $400-500 at the end.
2) go to a Vet Dermatologist and get on the right track immediately. They may charge $100-175 at your first appt, but they WILL get you the correct diagnosis. You can locate a vet in your area by going here:
acvd.org | Home | Veterinarians with specialized training in skin, ears, and allergy
It sounds like your other dog also has allergies, as evidenced by the paw licking and chewing. I'd get a good allergy shampoo (NOT dandruff!) like Duoxo...get it on Amazon, not from the vet. Also, I'd get them something like Natures Logic Venison kibble.