I usually face the same issues coming from sea level-
To help acclimatize beforehand, Horseshoe Meadows would be good or if you want to explore more, check out the Bristlecones Pines Forest east of Big Pine and spend the day walking around there. It's a pretty cool place and you can get in some easy walking above 10,000ft. If you wanted to do more hiking another option is check out Kearsarge Pass for an acclimatizing hike.
For your other question, for my successful summits, I've stayed in Lone Pine the night before, hiking up higher during the day (10-12k) in the day or two before. Personally anyway, I've found focusing on getting a good night sleep has worked better for me than staying high and not sleeping well.
Good luck!
-Rob
