Regarding starting up the MMWT trail before midnight on a day hike permit:
I thought this had been definitively answered before, but searching for it, I couldn't find anything.
So I emailed our contact in the Wilderness Permit office for Inyo National Forest, and this is the response.
Lone Pine Lake is not within the Mount Whitney Zone, but it is within the wilderness boundary. This means camping at Lone Pine Lake requires an overnight type wilderness permit for the Mt Whitney Trail. You can day hike to Lone Pine Lake without a permit.
A Day Use permit is valid to be in the Mt Whitney Zone from Midnight to Midnight on one calendar day. Technically you don't need the day hiking permit until you are entering the Mt Whitney Zone just above the lake, so you could start on the trail before midnight, timing your approach to enter the "Zone" after midnight. It would be ok to take a brief rest break before the zone boundary, but a lengthy rest break that involves sleeping might be considered camping.
Hope this answers the question.
Cheers