Biting is always a risk if an animal has teeth, but given how friendly the cat seems, I'd say you're not super high risk for a bite. He's asking for affection, and showing you some. If you want to try petting him, I'd suggest you be sitting/squatted down, more at his level. As he's coming up to you, slowly stretch out your hand, so he can sniff it (like how you would greet a dog you don't know). If he's as friendly as you're describing, most likely he'll rub his cheek on your fingers. He'll do the "work" and you hardly need to move your fingers at all. If he doesn't come up to your hand, or walks away without sniffing/rubbing it, then you can just leave him alone. He'll tell you, if you listen.
The worst thing you can do is be nervous. He'll pick up on your anxiety. If you're too afraid, perhaps someone else could make that first contact for you?
Do you have any "cat" friends? Anyone in your life who's good with cats and could come greet him for you? Maybe look him over? Sometimes just being confidently handled by someone who's not afraid and knows what they're doing can work wonders.
I foster lots, and so I constantly have "new" cats with unknown histories. Most of them very protective mothers with kittens and crazy hormones. Letting them sniff your hand, and come up to you on their own terms is what works well for me. Also, while I've been scratched some, I've never been bit. Generally cats don't go to "bite" as their first reaction. First reaction is usually "run away." If they can't, "growl" or "scratch" are still generally above "bite."
Regarding him not using the shelter:...first thing you have to know about cats is that they won't do something just because we want them to. In fact, if they know we want something, the frequently like do to just the opposite.
It's kind of funny and cute, except when it's your cat that won't behave.
A garage can be a safe place to live, if you pick up the things that are dangerous. In my garage, the chemicals and things cats shouldn't eat are kept on the top shelves. However, if this cat is truly a stray and has never used a litter box, you'll have to be prepared for there to be messes. If you won't deal with that, then locking him in the garage probably isn't the best idea. If you keep your car in there, make sure you locate the kitty always before you start it, as engines can kill cats if you turn it on while he's sleeping up there.
Keep offering water, but if he's been on his own a while, he probably has other sources. it's worth offering, but don't freak out if he won't drink it. You could try putting a little tuna or something else yummy smelling in the water to make it more enticing.