Hard to guess the weather in Colo lately due to changes in the Winter snows and lack thereof this year. "Normally" the time frame you mention is generally pretty good as the rains start toward the end of July and into August. I have been snowed on in every month of the year while in the Mts, but usually you get rain, some minor hail storms and no snow from early July to late August. I've never bothered to check the temperatures en route or on top.
You can pretty much camp anywhere along the hiking/climbing routes as long as you use good judgment in picking your camping site but I question why you would want to bother as most of the 14ers involve short hikes and carrying all your gear to camp overnight would hardly be worth the effort. Exceptions to this would be the longer trail approaches to 14er summits such as Pikes Peak, Longs Peak, Capitol Peak, the 3 in the Chicago Basin, the Blanca group in Lake Como area, and maybe Snowmass. Of the 55 14ers I have only backpacked for Capitol, the Chicago Basin 3 (which also normally involves the Durango-Silverton Railroad trip to the Basin trailhead) and the Blanca group up the terrible Lake Como road. Certainly for Elbert or Massive you would not want to backpack and overnight camp and you would car camp if you chose to drive to Kite Lake for the Democrat, Lincoln and Bross loop.
