Well back in the 13th century which I suppose DF is supposed to be approximately based on, most surgeons and dentists were actually barbers. They had the experience of doing delicate work with very sharp knives. Although some people claim that quite a few were also torturers, experts on just how and where to cut people without making them die.
DF does a reasonable job of this if you set it up that way; you could have a chief diagnostician who is your "academic doctor", and a separate surgeon who is more "working class" and has some related job (animal dissector, butcher, etc.). Most players don't do this of course.
Cutting up cadavers did not really come into the picture until a few centuries later. Even then it was done generally in secret since church did not like it at all.
That's tricky. I'd argue that it was more common earlier; Pope Boniface VIII (funny hat 1294 - 1303) had a variety of issues, including some odd ones about "bodily integrity". As I understand it, the papal bull of 1300 prohibiting
Mos Teutonicus (basically, turn corpse into skeleton for more practical transport home e.g. from a crusade) did not technically ban post-mortem dissection, but was apparently interpreted that way in some places, and had a chilling effect post-1300.
On another point, medieval people should be much more accustomed to seeing blood and death around with butchery being more or less daily or at least weekly business at a farm, though in DF butchery seems to be just pressing the magic button that makes an animal fall apart into neat stacks of meat and bones.
Hmm, I'd argue against "daily", and probably not even "weekly". But for traditional farmers not at either the lowest or upper levels, a few times a year certainly. An additional wrinkle is that in DF, farming is so much more efficient, and pasturing so iffy, that it's arguable whether peasants would have a "meat industry" at all. Raising poultry mostly for eggs is both historically valid and functional in DF, though, so you'd end up with at least some old birds "retired" to the soup pot.
That said... Butcher and Animal Dissector should train Discipline fairly effectively along with the medical skills; with Trapper, Ambusher, Fish Cleaner, Fish Dissector, Tanner, and (probably) Alchemist having at least some as well. Note that this is really two suggestions: one is that pre-generated dwarves with relevant skills should also be given a good chance for some Discipline; and also that training these skills should also train Discipline.
Assigning your battlefield-surgeon-in-training to a few months of sparring with a unit, and a few months of hacking up all sorts of bizarre animals for the kitchen, should be pretty good preparation for the brutal reality of DF hospital care; it makes both real-world and DF internal sense IMO.