I've done all except pure Stealth, which I tried, but it didn't work. Pure Mage can be done fairly easily. Combat is hard to keep "pure" as you need to heal and that requires at least potions, but limiting yourself to just potions you find can work out. But unless a Stealth character can start with 100% Chameleon, there's no way to avoid using magic or combat skills as well for defense. You also have to consider Alchemy a Sneaky skill to get much offense!
Yes, this is an area where the changes from Morrowind to Oblivion resulted in less choice. In Morrowind, we had Short Blade, a Stealth skill, so it was possible to have a viable pure Stealth character.
As for the healing issue, it's always possible to simply rest until one has healed, and not use any skill for that process at all.