I have had a similiar fungus infection of unknown origin as well, but it has not spread to my kids. I don't know if this is possible but I am almost certain I got this from contact with someone's saliva. I know it sounds crazy but let me explain. Last summer I was involved with someone. During an intimate moment, he deeply kissed my neck and also my forearm (opposite the elbow). Right after that a circular red raised rash appeared in both places where he had contact. At times it was itchy, especially after coming in contact with sweat. It would then get a ashy dry oI thought nothing of it, and it was thought to be attributed to using a public pool. Relationship ended a short time later. Fast forward to this summer. Reconnected with same person. Same set of circumstances and BOOM same darn rash now on both sides of the neck and both arms. What gives? On an upnote - after no success with anti fungal cremes, I used a product called No Miss Anti Fungal liquid drops (used by nail salons) it contains Miconizole and applied it with the dropper and a Q-tip directly to the rash. After it dried, I used the oil contained in a Phosphadityl Choline capsule over the area. The rash is finally starting to clearing up - rapidly!
"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