It does not survive off of the cat for very long so it doesn't live or breed in carpets.
But ringworm fungal infection can lurk in carpets and home to infect another pet.
You can read about that here
I'd suggest starting with asking another local vet for the cost of an office visit and a skin scrape. If its mange you can ask about using Revolution and they make that for dogs too which would clear your dog of mites if the dog has picked them up.
You can read about how ringworm is treated on the above site.
What you want to know when you call a vet is how much for an office visit, expected/estimated costs to diagnose a possible skin parasite to decide if you can manage the costs.
As cash flow is an issue Cher (jessesmom) did give you a link with lots of potential financial resources but you might also check with your local humane society about low cost vet care in the area or consider seeing if you qualify for Care Credit so you could do time payments for care
A credit card company for health care, including veterinary care.
"With a comprehensive range of plan options, for
treatment or procedure fees from $1 to over $25,000, we offer a plan
and a low monthly payment to fit comfortably into almost every
budget." They have a zero interest program if you pay within a certain time frame and they list the vets who work with their product.