I usually follow a slightly different path for Nightmare!.
The real trick to it is to get through each level as fast as possible, while using tricks to prevent resurrection *Killing on open doors, killing on water, killing over another corpse*. If you ever enter a level and it says its really humid, go find and use any levers until you cover the entire level in water, making resurrection nigh impossible, and allowing you to clear the entire level without any rez problems, *aka Slow and safely.* however there will be no you feel safe message.
Chaingun path is a must, but carrying around a combat shotgun for when alot of formers group together isn't a bad idea in the beginning *provided you find one*. Grab as many cells as you can, and a plasma rifle, and try to save it for phobos hell. A rocket launcher can be very useful as well, particularly if you need to clear out a real large room of foes. Try to save that combat shotgun and a few shells. If you find the City of Skulls, you have a good shot at clearing it with one, netting you a BFG.
For traits, I'd start out going real offensive with 1(2) EE then 3 SoB, then get defensive with ToN, Ironman, and/or Hellrunner. I wouldn't choose badass, you'll be getting into to many fights for anything but 3x badass to really be useful, and having either 3 extra armor points, 30 max hp/better medkit healing, or 30% faster mv speed/dodge is much better than no hp degrade over 100%.
Remember that powerups time stacks, and the time is doubled for N!, so take advantage of that. Use Berserk and Inv wisely, either to kill for exp, or when you have enough supplies to kill cybie, to get down there as quickly and safely as you can. I'd tell you to grab any unique if you see the message...but that's kind of a given. If you're doing really well though, it might be better to ignore it.
DON'T try to clear most special levels. But, entering one as a shortcut to the next level isn't a bad idea. Hell, you can just walk into Chained Court and grab 4+1 Berserks, or go to the Halls of Carnage for a quick supercharge *and you could go for that BFG as well*.
Last but not least, if you fail...destroy all evidence that you tried :P. Then try again. Usually it just takes that perfect game to beat Nightmare!, most definatly if you decided to do a challenge Nightmare!.