I don't know what the snowpack is like in the southern Sierras this year, but up here in the Coast Mountains of Canada and Washington it's very low. So the hiking season is likely to start quite early. If it's the same for the Whitney area then perhaps late June would be okay.
As for altitude sickness there isn't much to say. Check out the things
you should do to avoid it, but if you do get it on the hike then that's just the way things go and you'll have to deal with it if it happens. You won't be able to predict in advance whether you will get it, and I don't think there's much you will be able to do to prepare for it. Normally doing some other high-elevation hikes beforehand might help but it doesn't sound like you're going to be able to do that.
At any rate it sounds like you have the planning part under control. You'd be surprised what young and slightly crazy guys can do, so good luck with the project.