Do not apply soap on that. Com that it was caused by his acid reflux. Plus the cat remains contagious to anyone in the home. Anyone including young children and adults can suffer from atopic eczema. While ringworm is caused by a fungus, lyme disease is bacterial, so again, treatment for the two skin conditions is very different. You have answered all the questions.
Ringworm prevention: stay dry
Keep the areas dry and use anti fungal powders or sprays. I wish i had been even more cautious with her food intake and that i had been pushier with her pediatrician because she was suffering from these allergies for so long.