You just might need crampons and the ice axe, especially hiking through the night. The switchbacks between Trail Camp and Trail Crest might be useable by then, or maybe not. You should talk to people coming down and ask about them. Also watch for trip reports on the conditions just before you hike.
That's pretty ambitious to get to the summit from the bottom by sunrise. (A few people could make it.) You can start at 9 or 10 pm. You don't need any permit until you reach Lone Pine Lake, but after that you do.
