I'm the opposite of the others.....I'll give you some advice on the ringworm, but not the sleeping. Generally, I wouldn't recommend using peroxide on the rashy area. It is good to bubble dirt out of fresh wounds, but it can actually bubble away the cells that are promoting healing of the spot. Also, I always recommend going to Walmart foot care aisle and getting their "Antifungal Cream". It is clotrimazole cream 1%, which is the same medicine that is in Lotrimin. The best part is that you can get a 60 gram tube for about $5. You can use this cream twice a day, and you can alternate it with antibiotic ointment (like Polysporin). Topical antibiotics have anti-inflammatory properties, so they can be helpful in that respect and they won't worsen the problem. Neither the clotrimazole or the antibiotic ointment should irritate the area. If you notice any worsening of the area at all, your little one could be sensitive to some component in either medicine. I've not seen anyone sensitive to the clotrimazole, but I have seen many people that develop sensitivities to topical antibiotics.
Other than using those two types of topical medicines, you should try using soapless cleansers (like CeraVe or Cetaphil) for washing his face so that his skin keeps from getting dried out. Also, moisturize the area after washing his face (again, with products like CeraVe or Cetaphil) throughout the day after meals and at diaper changes. This will help his skin barrier to function better and want to rash up less.
Hope his rash clears soon!
A.