You are a good person to want to help how you can, but giving money for his baby momma to buy a car is a terrible idea.
I think it's fair for him to list both his child and sister as dependents for housing and taxes, since it sounds like they are and they will be living with him. Even if he does qualify, then that will take time.
It seems like housing is the big problem at the moment, and I understand that. Living close to either his work or her school is a good solution. I once paid moving expenses and 2 months rent for a loved one to get them into decent housing which they couldn't understand was affordable when really it was. Sometimes people just get warped ideas about things and in my case the person lost their previously crappy housing to a fire and was functionally homeless but wouldn't even look for a place to live. Plans were morphing into miserable couch surfing indefinitely. I had to do the looking and just say "I looked at about 10 places to pick the best fit and it looks like you should move here or here, one of these two-three specific places. Let's go look at them so you can pick. I will hire movers and pay the first two months rent, so no more excuses." It worked because it gave him some choice and feeling of control, but also took away his inertia and excuses.
Could you do something like that to get him, your aunt, and your sister moved to a better location? Can you help him by spending a day this weekend vetting new apartments, then offer to pay the first month or two of rent if he moves?