I don't think ONLY hanging is a good idea for reasons already mentioned. But if you're working the muscles in other ways then your ability to hang safely will increase. If you're hanging past the point where it's comfortable I'd be concerned that you were hanging off the ligaments. That said, if you're really so tight that your muscles won't stretch out to the point where the ligaments are taking the load then it could well be a good thing.
I do a lot of training on gymnastic rings and I don't really think it's particularly good for grip - it's easier to hold on to rings than a bar. For grip I HIGHLY recommend ropes. We have battle ropes at our gym and I rig them up on pullup bars so you have two parallel ropes hanging (one for each hand). Use them to hang, do pullups, hang upside down, skin the cat, levers....just use your imagination. It works wonders for the grip - those big, chunky ropes really get all the fingers working hard!
Now on rings OR ropes hanging upside down I think will always be good for you. It works the lower traps and to me feels very therapeutic. So the position you'd be in is as if you were standing with your hands by your sides - but upside down and holding onto the rings/rope. Works the core too!