It sounds like he's making an effort, at least. I've seen many people with pretty serious mental health problems get sober and stay that way, but they definitely have challenges the average alcoholic does not.
It's up to you whether he remains in your home. You can be loving and supportive while finding alternative living arrangements. There are places out there, like sober living houses, and programs for people with dual diagnoses.
His problems are no reflection on you, personally. So it sounds as if one of YOUR main challenges is to work on maintaining healthy boundaries with others, including him. If you choose to give him a gift (i.e., the car), it's his choice what to do with it. You can make a boundary that if he makes poor choices, he lives with the consequences of those choices (i.e., he buys a clunker that needs repairs, he pays for the repairs or finds alternative ways to get around). You don't need to be managing EVERYTHING for him. A bit of help with legitimate unexpected problems is one thing, but managing the consequences of his actions is another.