Haven't had one of these for awhile. This thread is for questions or comments that you've felt silly about not knowing/understanding. Let's try to exchange info that seems obvious, knowing that due to the illusion of transparency it really isn't so obvious!
To repair hangnails: Nail cream or nail oil. I had no idea these products existed, but they do, and they are designed specifically to deal with this problem, and do a very good job IME. Regular application for a few days fixes my problems.
To prevent it: Keep your hands protected outside (gloves). Minimise exposure of your hands to things that will strip water or oil from them (e.g. detergent, soap, solvents, nail varnish, nail varnish remover), and when you can't avoid those, use moisturiser afterwards to replace the lost oil.
(Explanation: Splitting/peeling nails is usually due to insufficient of oil or more rarely moisture. I've heard some people take a paleo line that we didn't need gloves and moisturiser and nail oil in the ancestral environment. Maybe, but we didn't wash our hands with detergent multiple times a day then either.)
It's not the nail itself, it's the skin around the nail...