Ok, sounds like you have a decent amount of time. The June lake area sounds like the best place for you. Oh ridge isn't all that bad. You can get reservations there so you can probably request a site near the few trees and get a windblock. When the weekend crowd leaves you can move and get pretty much any campsite in any of the campgrounds.

Take a ride up to the ghost town of Bodie, check out the hot springs at Hot Creek (bring your bathing suit if you want to hop in), all the lakes have pretty nice hiking. You can checkout Devils Post Pile in mammoth, but you have to pay and take a shuttle in. DPP is a rock formation in which the rocks are hexagonally shaped from volcanic action. Not sure if it's worth paying to see it, but if geology is your thing...

Yosemite is pretty close so you can make a trip out there on one of the weekdays when the crowds die down.

I hear the restaurant at convict lake is pretty good. On your way to vegas, be sure to stop at Schat's bakery and get some bread and pies. Best bakery ever. I wish I could tell you where the cheapest gas is, but it's out in the middle of nowhere on 395 and I couldn't even describe it.