Yes, much to digest! Here are some starting comments...
I was worried about your total group size. But 14 is certainly not too large a group.
As to the competitive part, the National Forest has a policy against contests. In the past, there were Badwater to Whitney summit contests, but now they stop at the Portal. But in your case, you only have two teams competing, so that would just be between members of your own party, so it isn't an open contest. I certainly wouldn't make the competition a large part of your publicity, though.
As for lodging, I am sure others will chime in. I've stayed at Doug's Whitney Portal Hostel in Lone Pine -- it is a true hostel setup, where you can rent by the bed, or an entire room. It is a clean and simple facility. Otherwise, you have a choice in a number of motels.
I would stress, for the people going to the summit, that they learn about the effects of the altitude, and the symptoms of AMS. Although some people are not affected at the 14,500 elevation, a majority are. Some are really hit hard just after 12,500. That being the case, I would highly recommend acclimatization, the best would be to stay two nights at Horseshoe Meadows (10,000'), a walk-in campground area about 35 minutes out of Lone Pine (Start up the Whitney Portal road, then turn south). There are some nice conditioning or warmup hiking opportunities there (Cottonwood Lakes, Cottonwood Pass, Trail Pass & Trail Peak)
