Which durable full-frame DSLR is best for large prints, document reproduction, video, and tethered shooting?

The choice is probably going to boil down to the 5D (Mk II or III) or the D800 (or D800E).

The 1Dx is a lovely camera, but the resolution is lower than the 5D2/3. It will do just about everything you want, and do it well, except that you'd really notice the missing pixels printing at 30x40. Even the 21.1MP of the 5D is going to be cutting it close in terms of acceptable resolution. The difference between them isn't huge by any means, falling into the just noticeable category, but there's no sense shelling out the extra money for something that is going to be marginally worse for at least two of your use cases (document copying and large prints).

For single-shot document copying, there's no contest: a D800E will win by such a large margin that it's unfair to enter the other cameras (even the non-E D800) into the race. If you go for a rostrum scan and stitch, though, all the extra resolution really gains you is fewer exposures required (which can be a huge advantage by itself if you're a conservator trying to minimize the document's exposure to light).

For that matter, the D800/D800E is just barely stepping into the territory where a 30x40 print might be considered "exhibition quality" at over 150 pixels per inch native, but it won't have anything like the almost tactile quality of an image generated on a high-resolution medium format camera/back. If you do everything just right, the "E" will visibly outperform the ordinary D800 (unless you have to invoke the moiré fixer in the included CaptureNX 2 version).

You can easily make pictures that look great from a reasonable viewing distance with a 35mm-format DSLR, but I've had openings and exhibitions, and you just can't keep people from getting up close and personal with the work without velvet ropes and security types. As long as you're not hanging your stuff beside somebody who's using an IQ180, P65+ (or their Leaf brethren) or an H4D-60, you should be okay. If you need the appearance of physically etched fine detail, though, you may want to rein in your print sizes just a touch—200ppi or higher is so much of a visible improvement over the 125-160ppi class that the prints, though smaller, will be all the more impressive for it. (That would equate to 24x for the D800 or 18-19x for the 5D; smaller still if you want a sort of daguerreotype tactility.) Upscaling the image can help somewhat, but there is no real substitute for captured pixels. Unfortunately, the air is rather rarefied in the hi-res medium format world, and low-light performance isn't great unless you bin away the resolution—and there's no video.

Between the 5D2/3 and the D800/800E, there isn't much to choose as far as low-light performance and video are concerned. Nikon is arguably better at the extremes, but we really are talking about the extremes—for static subjects, you really shouldn't have to worry about using the highest ISOs either of these cameras can use. There is enough of a difference between the 5D2 and the 5D3 in terms of autofocus, metering and overall handling to make the newer model worthwhile (and used 5D Mk IIs aren't exactly five a penny anyway). You'll need to take into account the fact that you'll need additional software to use the Nikon tethered (Camera Control Pro 2), while Canon includes the utility in the box. (I won't vouch for either; the only tethering software that I've ever really liked has been Hasselblad's Phocus, and that's mostly because it will do everything with a 'Blad except move the tripod for you.)

I'm sorry that's not really an answer, but I do hope that it helps to expand the question a little and get you pointed in the direction you really want to go. If you had an existing stock of glass or were familiar already with the Canon or Nikon ergonomic style (they're pretty much backwards compared to one another in most respects), I'd say go with what you know. Starting with a clean slate, though, the choice is all yours.