Why are there clothes on the dining room chair

Firstly, do you have a hospital? If not, build one, even if it is just a bed somewhere random. (Your dining hall would actually be a pretty good place)

Even though you don't have any dwarfs that are trained in those skills, it doesn't mean they can't do them. It just means they'll suck. So pick a dwarf who isn't busy and activate the medical skills on him and deactivate all the rest. He should run and get the dwarf and carry him back to the hospital bed.

Make sure you have buckets and prepared meals. Wounded dwarfs who can't feed themselves need to be feed by other dwarfs. Buckets provide drinks, and prepared meals provide food. I think just any food can be given to a wounded dwarf, but I've always had the most luck with prepared meals.

Finally, wait. Either the dwarf you esoterically assigned to be a doctor will patch your dwarf back together (assuming you have a traction bench, cast powder, and crutches) or he'll keep him alive until a more skilled dwarf comes to save the day.

If you just want to fix it (i.e., remove all the piles of vomit) you can download DFHack and use the "clean" subprogram to remove all contaminants on the map. Works on blood, dye, snow, vomit, and ichor, I think.

If you do that and your animals remain nauseous, you might have some underlying cause.