It turns out that my son and I will be backpacking and camping in the Grand Canyon May 14-16, and hiking out on the 17th. I will be doing the hike barefoot (or as much as I can manage). I'll be at Bright Angel, Cottonwood, then Bright Angel, and then out. I should be back at Mather Campground sometime on the afternoon of the 17th. (My campsite also turns out to be pretty close to your-alls.) Rick has suggested I stop in, and I will try to do so.
I did this last year about this time of year (but only one night down). Here are suggestions (keeping in mind I am a hiker and not much of a runner). It was really hot in the canyon: 104°. A trick it took me a while to glom onto (even though it
is in the suggestions) was to wet my shirt and hat, and re-wet it whenever possible. This time I might also consider a bandanna I can wet and put around my neck.
Make sure you check that there really is water all along the route (check in at the main visitor's center). They claim it is seasonal, and May is about the time they turn it back on for the higher elevations.
Heading down the South Kaibab should be fine (it's also the shortest way down). The buses to get to the trailhead run every 30 minutes from 4:30am to 6:30am, and every 15 minutes after that. An early start is good for avoiding the heat as much as possible. (I found the hottest stretch to be along the Colorado River.) When I went down before I was passed by a lot of runners (fairly early on), so I'm sure they had no problem hitting the bottom and coming right back up. They were all shod, though.
Be aware that the sand really heats up, and at some point (around 10:30am) gets too hot for bare feet. Also, regarding bare feet, Phantom Ranch has a sign requiring footwear. I'll be writing them in advance to see if I can't get that removed, but based on past experience, these folks never change their minds and make up every excuse in the book to justify their beliefs.
While the buses have no rules against bare feet, I (and other folks I know) have had trouble with a few (just a few) of the bus drivers, and management backs them up. (In this case "management" means the bus company contractors, not necessarily the NPS.) I will also be making an effort to guarantee no hassles with the buses, too, but who know how that will turn out.
http://www.nps.gov/grca/planyourvisit/upload/Corridor_Brochure.pdf
Be aware that if you are really doing a Rim-to-Rim-to-Rim you are talking 44.5 miles (going down South Kaibab, up and down North Kaibab, and then up Bright Angel). I am such a wimp I am not even going all the way to the top of the North Rim, and I'm taking three days with camping rests!
Cell phone service is provided by Verizon and Alltel. My company is AT&T and I had service, so it was obviously piggybacking off one of the others. I had no service in Mather Campground itself, but had somewhat weak service at Park Headquarers and pretty good service at the Bright Angel Trailhead. In fact, since my watch had broken, I was using my cell phone as my clock, and my wife managed to text me halfway up the Bright Angel Trail (above Indian Garden). Freaked me out to have my phone bing in the middle of nowhere.
Hope folks find all this useful.