Hi!
Red wounds on dwarves will heal eventually, but they will take a lot of time (after a long rest, they will become yellow wounds, which are also bad and only later become minor wounds).
Amputations (dark grey, I think) will never heal and cause serious trouble for the dwarf (phantom pain causing them to fall unconscious from time to time for the rest of their lives). Serious wounds to the nervous system will also never heal.
The job a dwarf has enabled is called "Healthcare", but when you check what the dwarf is doing, it will list "Recover wounded", "Give Water", or "Give Food" depending on what the dwarf is actually doing.
All dwarves have "Healthcare" activated as a default (even peasants), so if you never deactivated it, any idle dwarf should have it enabled.
As for permanently injured dwarves, it depends on how you wish to play. A lot of players seem to consider the inhabitants of their fortresses as playthings and will simply kill off any plaything that is no longer fun.
If you do want to go about it more positively, first you need to watch whether the dwarf falls unconscious from time to time (very probable). If that is the case and you have traps in your fortress (weapon traps, stonefall traps, cage traps), I recommend you deactivate all hauling jobs for that dwarf and give him jobs that do not have him walk through trapped areas (keep in mind that he will also walk to any meeting areas, dining rooms, and barracks/private bed room he may be using). Jobs that finish quickly are preferable, of course, as unconsciousness would otherwise interrupt them (so, unless the dwarf is very strong or a really experienced smith or stone crafter, metal smithing or stone crafting are not really recommended as hauling the material for the job would take too long).
If you have such a dwarf, you may also need to turn off "All dwarves harvest" in the options if harvesting (going to the fields and then bringing the fruits to the food stockpiles) were to cause the dwarf to walk through traps.
Deathworks