Mammoth almost always has skiing until July 4th, although sometimes it is just a few token runs.
The Shady rest campground is almost downtown, and more than a thousand feet lower than the main lodge. Good central location if you want to head into town for whatever, but not particularly scenic.
The campgrounds up the road towards the lakes are much more scenic, if they are open. A little higher and colder, of course.
Lots of camping down at Devil's Postpile, but private car traffic is limited on that road. During certain hours, the only way in or out is by shuttle bus, (with some exceptions, I believe, like if you are towing a stock trailer maybe).
Head out of town and there is a ton of camping on the June Lake loop, down at Convict Lake, Lower Rock Creek, etc.
Take a look at the USFS website and look for camping links on the eastern Sierra.