cborigo: Your trips to the Whites ought to be all the indication you need of what you need to be prepared for -- everything and anything -- unless they have all been to the huts in late July.
I wouldn't even think in terms of "average" or "typical". As far as weather related conditions, think of this as doing Mt. Wash up Tucks and the headwall, off trail, at about the same time of year. How would you prepare for that? When I do the Whites, I pretty much load everything I can think of in the truck and do my final packing at the trailhead.
Weather patterns in the Sierra are somewhat more predictable -- short term!
-- than in the Whites, but that slight advantage is offset by the limitations on what you can bring from PA and Vegas.
I would be prepared for all of the above, take as much as you reasonably can to cover the range of possibilities, and make your final cut based on the absolute latest report from the above source. Depending on how you are traveling and whether and where you stay in Lone Pine, you may be able to stash extra gear in town. Many places -- Hostel, Dow Villa -- are very hiker/climber hip and I would be surprised if they could not accommodate some short term storage, especially if you are staying after the climb. Which BTW I [/i]highly[i]
BTW, as you also asked about route conditions, you should be aware that strictly speaking, the MR is only the Chute and the Final 400, and the usual approach is the North Fork etc to UBSL and Iceberg Lake. The point is that this is a route only, not a trail as such, and it takes some research to be able to find it in the best of conditions. Lots of info on this site about that, including video, so take advantage of that and be prepared for most of it to be under snow.