I'd say that there should be both spells that heal over time and ones that heal instantly. Heal over time spells could be if you want regeneration over time, while instant healing spells are used if you just need to quickly restore health.
Potions, I'd say, should always heal over time, though some could heal more quickly than others. In general, quicker healing potions should be more expensive and less common than once that heal more slowly, thus making them best saved emergencies, which is where you'd most need them anyway. By ensuring that potions heal over time, they can hopefully introduce some distinction between healing potions and spells, as opposed to having potions just be spells in a bottle. Although home-made potions already do heal over time, which is nice, but I'd like to see some longer lasting, slower healing premade potions too. I also wouldn't mind having control over the duration of the potions made using alchemy, with the catch being that potions with a longer duration have less strength, though if such a system were limited, there should probably be a lower limit to prevent players from creating potions that are effectively instantanious.
Sleeping should heal you over time like in Morrowind, because it was kind of silly that you can fully heal all wounds with one hour of sleep in Oblivion.
If there is something like hardcoe mode in Skyrim, I'd say it shouldn't effect healing at all, except for maybe how sleep factors into the equation as sleeping would be mandatory with it active, so the game would not need to rely on healing you when you sleep to encourage players to sleep on occassion in the game. It's always annoyed me in New Vegas that I can't have my stimpaks heal over time without needing to deal with the need to eat and drink. Healing over time instead of instantly could actually add some meaningful challenge and strategy to the game. Needing to eat and drink is just boring and adds nothing except an annoying, tedious chore I already have to deal with every day in real life to the game.
I think it'd be cool if you didn't really have a "health bar", instead, certain races/builds can have slightly more threshold for pain/damage, but it's more about dodging/blocking/armor (but only strong people can wear heavy armor and still function, so the balance isn't broken!).
No, that wouldn't be "cool", I'd say it would be pointless. Sometimes, if a system works fine, you should just stick to it, unless you can come up with something better, and there's already too many shooters trying to do away with health bars, I don't want my RPGs to to it too.