My doctor can't fix him because he's dead,
Give all the medical labours to everybody. The individual jobs will be slower, but they'll be taken more quickly.
I'm running out of coffins
Make some more.
and I can't harvest any new crops because my farmers are running around babbling.
Allow all dwarves to harvest, using the standing
My metalsmith punched my fisherdwarf into the well (ironically) so there's no water to be had
Sure there is, it's just a bit bloody.
my hammerer went insane and clubbed all my brewers to death
Enable the Brewing labour on anyone with a non-essential profession.
So...yeah, how do I save my fort?
Shut down everything that isn't essential. If you have plants to spare, the current ones can be left to rot in the field. If you have booze to spare (say, six months' worth), you don't need to do any brewing. There is little else you need to do beyond getting the dead in coffins, or behind a door. I've seen dwarves bury a dozen teeth before picking up the corpse, so it may be a safer bet to have everybody dumping body parts in a holding area near the built coffins.
No hauling except for corpses, body parts, and perishable food. See below for another exception.
Do you have a jail, or any form of justice? If there are few dwarves sentenced to beatings, you can build an open jail by spacing chains so that no dwarf can attack a neighbour. Once half the fortress is chained up, the other half (who, not having flipped out and broken something, are relatively stable) can work on stockpiling food and drink in the jail area. They're always much more interested in hauling stuff than feeding and watering their fellow dwarves, and chained dwarves can pick up food and drink themselves. Everybody's happy! You can also put beds in the jail and make them unassigned bedrooms, and the chained dwarves will claim them and feel happy about having their own bedroom. Said bedroom may actually be pretty valuable, if it encompasses other beds and chains.
A mist generator is a good investment, though it has to be in a place that all dwarves pass through. If you build it to mist the jail area and store all the booze there, that should work out pretty well.