I have a few 4 inch seedlings with what I think may be damping off. At the base of the stalk the white root hairs disappear for about 1/4 inch at the base of the plant. If I place the plant outside during the day for some sunlight a small breeze will knock it over. If I prop it up and leave it inside under the lights it gets a little strength back but it is still weak.

Can I plant it deeply in a larger pot? If planted deeply it will stand on its own. If I bury about an inch of the root hairs will they grow roots and will I end up with a healthy plant or will it always be compromised?

The ONLY chance that your seedlings have is to plant them deeply and hope that they root before they succumb to the disease. This means ASAP because this one moves fast. I still would get them outdoors for some fresh air in the shade. Being inside just increases your chances of the disease progressing. However, please be aware that your chances of saving them are quite slim.

I have never had a damping off problem but this year I saw some white fuzz at the base of my new seedlings at the soil level. I was shocked and thought that maybe there had been some fungal spores in the potting mix so I sprayed the stems with a weak solution of hydrogen peroxide. The fuzz disappeared and never came back. The plants are fine so I don't know what it was but I don't think that it was the damping off fungus or the plants would have quickly died.

New seedlings under lights always have weak stems and can be blown over by the slightest breeze but they gain strength as they grow and are exposed to some movement. Some people use a fan but I put mine outside briefly for longer and longer periods and they get stronger very quickly.