I did the JMT Aug 10-24. It was perfect timing- perfect weather, few mosquitos, no mud, dry shoes, plenty of drinking water. Some days I saw lots of people, others I saw only 3 or 4. For the best weather, trail, and mosquito conditions I suggest going as late as possible within your time constraints (but still leave a few days to rest up before the friends have to start school).
Logistics seem harder than they are. I took Amtrak bus/train/bus from LA to Yosemite. After the trip I hitched a ride in Lone Pine from Whitney hikers to LAX. The bus dropped me in Yosemite at 8:00 PM and I made my way to the backpacker's campground behind North Pines car camp campground. I woke up early, got in line at the wilderness permit office and was on the trail at 9:00 AM.
For food resupply I started with 2 days food, mailed a resupply to the Tuolumne post office for the next 6 days, and mailed a 5 gallon bucket to Muir Trail Ranch for the final 7 days. I used a Garcia bear can with an O.P.Sack for the overflow. I strongly suggest renting the Bearikade Expedition so you can actually fit all food and toiletries inside. I took too much food- about 3500 calories per day. Next time I would plan 2500 calories the first week and 3000 the second. I'd also mail some stuff to Red's Meadow or just shop in the store there to cut down weight carried.
15 days was a good number- challenging but doable. I started out fairly easy with 8 days for the first half and was in great shape for the harder second (south) half. I started without hiking poles but was very glad when someone loaned me a pair at Muir Trail Ranch for the south half. My only physical complaints were sore feet early on and tightness in my achilles tendons. All of my training was walking 5 miles a day on city streets so 12-20 miles per day on rocky trails beat my feet up a bit at first.
If I really wanted to push it I think 11 days would be the absolute minimum for this 43 year old to do the trail. I would need to do a lot of training though. Taking it very easy I could see stretching to three weeks, but still doing the south half on one food resupply so no more than 10 days.