I would think you would need crampons for your hike to Lone Pine Peak, especially if you travel early and late when the snow is icy. Moosetracks went up the Meysan Lakes trail last year around the 10th of May. See this photo album
http://www.flickr.com/photos/moosepics621/sets/72157605016240554/Based on this photo, Last year on May 11th the northwest chute approach to Lone Pine Peak was already drying out, except for the bottom section visible in the photo
http://www.flickr.com/photos/moosepics621/2485940041/in/set-72157605016240554/As far as camping goes, you could camp at Lone Pine Campground, Tuttle creek campground, or perhaps in the Alabama Hills.