First, watch out for the weather! Keep an eye out for possible thunderstorms anywhere in the drainage area of the canyon you are in. Flash floods are often caused by storms that you don't see or hear. If lots of storms are predicted, postpone your trip. When hiking, keep a quick escape route to high ground in mind (preferably out of the canyon, but at least 3-5 meters above the canyon floor).
Now, on to the nuts and bolts. Some canyons (Peek-a-boo comes to mind) are so narrow in places I cannot walk through upright, even sideways. There were a number of spots in the canyons you mention where I had to lie down and crawl through the wider areas closer to the ground and pull my gear after me. Plus, my kit is really small; a lumbar pack, a small tripod, a separate filmholder pouch and a vest for accessories that I can take off and drag behind me. A wider-than-your-ribcage-is-thick backpack may not make it through the tighter places.
I use a folding field camera, and a smaller one. A bulkier folder (e.g., Zone VI or larger Shen Hao) or a long-railed monorail would not be my choice in the slots.
Setting up the tripod and camera in these close quarters can be challenging. I like a tripod with short legs when collapsed that can be extended out 90° from the center line. Often, I've set up with two legs against a canyon wall and only one on the canyon floor. Also, I often end up with the ground glass really close to a wall, making direct viewing difficult if not impossible. A mechanics mirror on a handle can really come in handy at times.
As for lens choice: my most used lenses are the wider ones, but sometimes a long shot works out as well. I usually carry a 75mm, 90mm, 135mm and 203mm (210mm) when in the slots. If I have room I'll take my Fuji A 240mm too.
As to getting everything in focus. If you have both canyon walls in a shot, then your movement choices will be limited. Still, imagining where the plane of sharp focus fits in the scene will help you in finding the right movements. I measure total focus spread on the camera bed/rail as a check; the smallest spread between near-far elements in the scene = the optimum positioning of the plane of sharp focus. Still, I find that my set up is very close to one wall that I want in focus and stopping down a lot is often necessary, even with optimum movements. I base my aperture on the focus spread. If you aren't aware of this method, see here
http://www.largeformatphotography.info/fstop.html .
I find that composing an image with a viewing filter before setting up the camera to be very, very, very (did I say, "very") helpful when working in close quarters and with long exposures. First, I never set up without having a shot already. Second, I've already decided on my exact camera position and can set up the tripod once and then not move it. Third, while one shot is exposing (20+minute exposures are not uncommon for me), I can roam the canyon a bit and find the next set-up.
Depending on the time of year you go, the canyons can be quite busy. Setting up your camera for a long exposure in the middle of a highway of canyoners and effectively blocking everyone's passage is not going to win you a popularity contest, and your gear is likely to get bumped during exposures, etc., etc. If the traffic is high, you might want to consider the quick shots and/or coming back another day.
I've not found dust to be a problem in the canyons; but, if it's windy on the surface, dust and other debris can get blown in. Using your darkcloth to protect your set-up camera is a good idea.
Finally, and probably most importantly: The Southwest Slot Canyons are Holy Places; not only to the first peoples who live there, but also to many of us who have been lucky enough to experience them and the spirits that reside there. Treat them with utmost care and respect; they are priceless resources and unique and wonderful landscapes. The sandstone walls are fragile. Damaging a canyon intentionally is a sacrilege and a crime. Damaging one accidentally is bad Karma. Don't climb on or otherwise scratch the walls, don't move rocks, don't move anything. We are guests in these special places and our host(s) have done the interior decorating as they desire. Photograph these with reverence and respect and the spirits might let you come out with a meaningful image.
Best,
Doremus
www.DoremusScudder.com
). So I'm expecting some pretty long exposure time, especially with added reciprocity failure time.. Also, should I use a warming filter with the Velvia? 
)
Save the LF for the gulches, not the slots, Antelope perhaps being an exception. You'll have to explain the use of a blue filter.