Not sure. It's difficult to say. If he was burned and then developed cellulitis, he would require antibiotics. If he healed from the burn and now has a rash of the newly healed skin, gentle cleansing and non-comedogenic moisturizers might be appropriate. It should be looked at by a surgeon who treats burns before beginning treatment.
Depends. Treatment depends on a diagnosis and there are many causes of rash, but this is likely due to over moisturizing of the normal skin around the burn or a fungal rash from use of the antibacterial ointment. Either way try to keep the normal skin more dry and it may resolve.
Thermal or rug burn? It depends on the type of burn and the depth. Topical antibiotic over the counter antibiotic, if not allergic, may be all that is required. If deeper medical attention should be obtained, .