fit2climb's recommendations are right on...

You don't need an ice axe. Might need microspikes, but might not. I climbed earlier this week only in hiking boots, but there were a few icy spots on the trail.

My method is to sleep the first night at Horseshoe Meadows-- either on the ground or in your car (if it is big enough to lie down inside). Then second night, sleep at the Whitney Portal -- walk-in camp sites, or in your car. If renting a car with that much room doesn't work, then spending as much time high up and sleeping in Lone Pine is the next best option.

If you can drive up to Horseshoe Meadows and spend some time hiking on 5/21, Trail Peak a good quick hike. I think I did it one time starting at 3 PM, but got back to the parking lot in the dark. Here's a link: Trail Peak: Warmup / acclimatization hike

On 5/22, you could drive up to Whitney Portal and hike up to Lone Pine Lake, and see the part of the trail that you would be hiking in the dark. Here's a good list of other warmup hikes: Whitney warmup hikes

When you do the full hike, don't start out with too much water. Take a filter (or do like I do, just dip and drink), and get water from the streams along the way. There are streams at 1 mile (North Fork Lone Pine Creek), and second just before Lone Pine Lake, a spring at Bighorn Park (just before Outpost Camp), and a stream leaving Mirror Lake, then Trailside Meadow, and Trail Camp. Final water is at switchback 25 (of the 99 switchbacks). The only water I wouldn't dip and drink are Trail Camp pond and a pond just below Trail Camp. You should take 3 liters up the 99 switchbacks: 1 for the switchbacks, hide one at Trail Crest for the way down, and drink the third up to the summit from Trail Crest and back.