I would sleep at Whitney Portal first night or Horseshoe Meadows, rather than down in Lone Pine. Sleeping at altitude (moderate altitudes like the Sierra) is best for acclimatization. There are walk-in hiker camping spots just below the trailhead that are usually not full.
Not sure how susceptible you are to AMS, or how fast you hike, but you could even do this: Spend two nights at Horseshoe Meadows (which is 10,000' elev), and cut out the day packing from Outpost to Trail Camp. First hiking day, hike to Outpost spend night 3. Summit and back to Outpost. After summiting, spend another night at Outpost or hike out. Getting as high as you can and sleeping, resting, and taking it easy is the best way to get your body to acclimate.
2. You can mix Diamox and Ginko B. Studies have concluded that taking Ginko Biloba has no effect.
