Hydrogen peroxide and iodine solution are both great for minor infections, cuts and scratches. Go ahead and treat the cat with what you have. Neosporin is also great!!
Is the cat neutered? Why would a fixed cat get into fights unless he wants to mate with a female? It sounds to me his hormones are driving him a bit crazy and he wants a girlfriend. Some cats, even after being neutered or spayed can either go in heat for a female or look for a girlfriend. Either way that cat should be kept inside all the time. If he cries at the door your mom should spray him with water. He'll get it. That's how I discipline my cat.
There's a product called "Fungiside" you can get at the pet store (or one similar). It's excellent for preventing infection.
however, this cat should be kept inside. Declawed cats have NO real defense.
I have found that the stuff they give you when you get your ears pierced is helpful. If it is bleeding i put flour on my kitties booboos. Not alot just a dash of it. But you may be able to get something from the vet to clean it. I don't think it costs to much. Peroxide not so sure. Hope this helps.