"Rinworm" is the lay term for skin fungus--no worms involved.
If your family's rashes are all related--likely but not certain--the probable cause would be fungus caught from the dog. Most fungus comes from soil, not pets, but animal fungi can be more inflamed and harder to treat.
Harder, but certainly not impossible.
First, each of you (and the dog) should have scrapings and cultures done to establish the diagnosis.
If strong, prescription antifungal creams don;t clear things up, oral medications (Lamisil, griseofulvin) are safe and effective.
Skin fungus has no internal health consequences. Once you get rid of it, and you will, it'll just be a memory.
Best.
Dr. Rockoff