Red, hot and swollen is a good indicator of infection, and yes your toe can get infected even without a toenail to complicate things.
Make a doctor's appointment. Until then, soak that toe in hot water with epsom salts in it. That will help draw the infection out and won't cost very much. And it doesn't hurt. Soak it twice a day and let it dry thoroughly, but do not wash the salt off the skin, just let it dry.
Ingrown toenails are a terrible thing to deal with, but soaking like this and keeping them filed or sanded down REALLY thin can help. Filing keeps them from getting strong enough to dig in.