Welcome, Camminatore!
Regarding the midnight start: I have been told by National Forest staff that you can hike up to Lone Pine Lake without a permit. It is only beyond the lake, and the Wilderness boundary, that you must have a valid permit. So I suppose you could start at 11 p.m.
But that would be foolish -- the sleep deprivation will catch up to you before you reach the summit. And the summit is only half way! Lots more people leave at 3 or 4 a.m. You won't be alone.
All the info about acclimatization is on the mark. Here are two recent discussions worth reading:
Anyone know about a High altitue B&B or Campground? recomendations for an acclimatization route up the summit? 
