I hiked it in Sept. 2007 and I ran into people hiking without a bear can and using only campsites that had boxes, but I don't know if they were supposed to do it that way. There are quite a few along the way. Off the top of my head, I know that in 2007 there were boxes at Bearpaw, Lake Hamilton, Moraine Lake, Kern Hot Springs, Junction Meadow, also near the junction with the JMT. I don't know if there were any at Crabtree because I didn't leave the trail to check it out, but I know there is NOT a bear box at Guitar Lake, and there aren't any on the east side of Whitney on the way down to the portal. Assuming you confirm with SEKI officials that it's ok to do it that way, your idea of taking less cans than you need and using bear boxes along the way until the last night or two could work. Just be sure that SEKI is okay with it and be prepared to stop where the boxes are, which means near other people.
Regarding campsites being crowded, I didn't stop at Bearpaw, so not sure. Hamilton Lake was crowded but absolutely beautiful and is a good place to stop before going over the Great Western Divide through Kaweah Gap. The Nine Lakes Basin on the other side of the gap at the top of Big Arroyo is awesome, but no bear boxes. The timing on whether to camp there depend on how far you go the first day. Moraine Lake would be good stopping point after that or on to Kern Hot Springs.
I spent the second night at Kern Hot Springs, which was empty of campers, and you need to at least stop there for a hot soak, which is amazing after hiking so long.
Guitar Lake was more crowded with JMTers, but still a great place to stop and is spread out enough to not feel crowded (unlike Lake Hamilton.)
Caveat to all this is that I went after labor day, so I would think you might run into more people.
Re clothing, even in September it was plenty warm. I used a long sleeve base layer and carried a micro puff insulating layer. I had a lighweight baselayer pant and was fine. I had a 15 degree bag and was generally too warm except at the higher altitude at Guitar Lake. If you go in July it could be colder at Guitar than in Sept.
BTW - You have picked an awesome trail for your first trip to the Sierra. You will love it.
If you want to get excited, here's the link to our photo essay from our trip, but don't look too closely if you want to have surprises instead.
Let me know if you need any other info, and enjoy your trip.
-I should clarify the pants I brought for night were Patagonia lighweight capline and then I had lightweight REI Sahara pants for hiking during the day.