Sorry about the wall of text, I've been thinking about this a lot, in case you can't tell

. It's a bit more broad than just first aid and non-magical healing, but bear with me.
Good ideas so far, but I think there's some other stuff that hasn't been mentioned yet.
In terms of long-term care I think it would be good if you had different types of items suited to a specific healing purpose. Potions that heal wounds or speed healing don't really make sense medically, at least until magic is implemented. Convential potions (extracted from herbs) should probably be basically limited to deadening pain and / or treating poison or some illnesses. Suitures (stitches) would decrease bleeding significantly, but probably give your dwarf a REALLY bad thought , a la "Urist McInjured has experienced excruciating pain recently" depending on their toughness, unless you prevent pain in other ways, or have a legendary doctor dwarf. You could have gauze that would decrease bleeding and help prevent infection, but if you had an unskilled doctor it might increase the chance of infection, to say nothing of if it wasn't sterile or even sanitary (same applies for stitches). If you applied gauze your doctor dwarf could also have the option of using a salve or tincture that would speed healing, counteract infection, or deaden pain. You should also probably have some alchemical mixture that could be administered to dwarves to knock them out if your alchemist was skilled enough, preventing bad thoughts from having stitches put in, or getting bad thoughts from pain. It would probably have bad side effects or be extremely addictive (not to mention poisonous or deadly) if your alchemist dwarf was unskilled, but if he was skilled it could be quite useful to have the dwarven equivalent of ether. Even if you had such wonder drugs, surgery's still probably a stretch within the Dwarf Fortress setting, or at least surgery that does more good then harm, but might be a possibility for healing internal wounds faster if your doctor dwarf was legendary and you had said concoction and reasonable sanitation.
In terms of immediate care, Medic dwarves with the healthcare labor could also probably run out to the battlefield to drag back dwarves and give some medical care to the wounded, including applying bandages and setting broken bones before the doctor dwarves could treat them. It's also probably a good idea for adventurers to be able to do basic first aid or possibly even recruit a medic for particularly large groups of comrades. This is assuming, however, that DF players treat their comainions as anything other than meat shields

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Also, If dwarven hygiene is ever implemented if your doctor didn't wash his hands he might increase the risk of infection, too, and possibly make everything worse. This is to say nothing of having your dwarves realize that they can decrease the risk of disease by being sanitary. This of course could vary in degree, as dwarves washing the blood and vomit off their faces is a different thing than washing their hands after going to the bathroom or taking baths regularly. Of course, dwarves could be just so tough that unless they're up to their eyeballs in filth they won't tend to get sick. Having dwarves be disease-resistant might even help to balance dwarves against humans (fast breeding) and elves (immortal) in world generation. How sanitary dwarves will be is really a thematic and mood issue (not to mention realism and period-correctness) anyway and up to Toady.
A related issue is how in-period we're willing or wanting to be in terms of Dwarven medical knowledge. If we were going completely in period dwarves would think that disease was caused by bad smells - "miasma" and that putting pus into wounds would help them heal faster. Your dwarves would be bled or given copious amounts of laxitives in a feeble attempt to counteract infections, and sanitation, even in medical practice, would be unheard of. Yuck

. I think that because middle ages medicine was
just that horrible we should be less strict. Sorry middle-ages history sticklers

. Somewhat useful medicine isn't really that much of a stretch, anyway, as classical civilizations, (Greeks and Romans) as well as other ancient and period cultures outside of Europe had pretty good medicine. Some scraps of knowledge could have survived, and this also creates the rather amusing situation of searching ancient ruins or trading for medical advice. However, the validity of that advice would of course depend on what philosopher you listened to, as some had some
really terrible ideas, and the lack of scientific method in that time period let them run wild. Even with that said, the hippocratic oath- to first do no harm- came from ancient times, and is sworn even in modern medical practice. So, as long as you don't assume a horrible dark age followed the fall of all civilization in every generated world, you have quite a bit of wiggle room in terms of what medicine can really do, even in a "medieval fantasy" setting. In my humble opinion it would be okay to gloss over how horrible medicine really was in a historically accurate setting in the pursuit of making the game fun.
However, the complete opposite viewpoint could be true, and what people
thought cured disease in period actually did help in the Dwarf Fortress universe. For example, giving people magic talismans to ward off miasmatic fumes would actually be a valid public health strategy, bleeding people or using leeches would cure diseases, and soap wouldn't really help to prevent disease. This would be a pretty interesting perspective, but in the long term I think the best road would be if
both treatments were essentially valid to some extent after magic is implemented. This could lead to awesome things involving reality-fantasy crossover like being able to use magic spells to sanitize a wound or prevent pain, having only really legendary doctors/alchemists/biomancers or artifact medicine able to heal wounds directly, or have strange moods that could create magic golem-like or mechanical prosthetic limbs or organs that would restore (or possibly even enhance) the abilities of your dwarves.
I think a innovative and cool mix of realistic medicine and fantasy healing magic would be awesome, and I think this really fits the mood of Dwarf Fortress. This isn't to say you need 19th century medicine and throw period out completely, but it would be cool if doctors/ healers blurred the lines between magic and medicine, with it actually working. Even though some sort of weird industrial-age steampunk setting would be cool, that's not what I'm suggesting as I don't think that's really in the mood of Dwarf Fortress in the near future.
Yeah, that was kind of all over the place

. Hopefully that made some sense to some people. I hope the pain meds aren't messing with my brain too much, as I just got my wisdom teeth out

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