Stop giving him honey. It is not pasteurized and is not recommended for human infants, so I don't see where it would be good for a young puppy (from your other thread). Not good for any compromised immune system...
I will pray for you and your pup, I can't imagine what you're going through. Have they checked you out for any type of parasites? I would look into hyperbaric oxygen treatment and look into taking coconut oil (good for pups too).
And I hate to ask this, but is there any chance he has picked up something from you? Not trying to be rude or mean... just find it an odd coincidence that your cat was affected too, and everything environmental seems to have been eliminated (by moving). Do you have a relative or a friend you could temporarily rehome him with for a few weeks or so to see if his problems improve any?
If that doesn't work, I hate to suggest this too, but contact that Last chance animal rescue group out west,
Best Friends Animal Society Home Page ... I've heard of two dogs that had untreatable allergies in the East US, but when they were sent out west, just the move to a different area helped them. They couldn't find out exactly what the dogs were allergic to, but the change to a drier climate helped them immensely.
And I would really think about not having a pet until your health problems are under control... there are things that we can pick up from them and until the doctors know what is going on with you, IMHO it would be safer for you and them...
Again, no harm intended at all, and I will keep you and your pup in my prayers for a quick, definitive diagnosis and even quicker recovery.
Love and Hugs, Kathy