What do I think your dog has? An incompetent or indifferent vet.
Panting can be due to overheating, exertion, illness, anxiety, pain ... not wanting or able to stand, eat, or drink indicates that the dog feels very ill. Leaning to one side does suggest a balance problem. As others have mentioned, could be vestibular disease, or it could be something as easy-fix as an ear infection (which can make a dog go off-balance and make him feel pretty darn sick, too). It could also be something as serious and scary as a malignant brain tumor. Sorry, didn't mean to frighten you, and I sure hope it's nothing like that, but some dogs do develop cancer and it can be managed if not cured.
Did your vet do ANY testing? I find it hard to believe that a vet would just send him home with a cheery
'Hope he feels better' and no diagnosis or treatment plan. If your vet really did that, I think you need to find another vet who will listen to your concerns and take into account all the symptoms you've observed. I'm not a vet, but from your description I am very concerned that your dog feels unwell and isn't getting the treatment he needs to feel -- and get -- better.