People like this are unlikely to do any follow up and will only seek medical assistance when it gets out of control. If the man is a veteran, he can be referred to the VA. Otherwise, there are other options if he chooses to pursue them. Yes, he needs to be check for all STD's, including Hepatitis, GC, chlamydia, and TB as well as HIV and syphilis. He very well could also be diabetic with his weight, life style and open wounds. He also probably has a multitude of other issues. Many homeless have histories of ETOH and illegal drug use. You already mentioned mental illness, and yes, that is a huge problem with this population. Which came first the chicken or the egg??
He clearly would benefit from a full checkup, labs, chest x-ray, wound care referral and a med regime. There are a lot of meds that are provided free via health departments or other social services as well as the $4.00 list at many pharmacies. All that said, he is still homeless. He could probably find shelters on cold nights and others if he is sober and well behaved. That is often a big IF.
Mental health options are horribly limited in this country if you have no money. I suspect this man will continue in his unhealthy ways until he dies of some horrible wound or is killed on the street. Even if you could get this man cleaned up, diagnosed and placed on meds, he is very unlikely to comply. It's a nice idea, but it seldom works.
Good luck, and welcome to the sad facts of life.