Will I need a permit to camp and hike around Horseshoe Meadows, Cottonwood Lakes, etc.? Do the rangers ever allow anyone to camp there more than one night? Also, are the camp sites at and around Horseshow Meadows near the parking area and the bear boxes, or are they spread out along the trail?
You only need a permit to hike in this area if you will spending overnite in the field. you can spend several days in the backcountry, if permitted for that.
As for the trailhead "backpackers campground", there are two, located at both Cottonwood Lakes, and Cottonwood Pass trailheads. Each has a one-nite limit, although I suppose it is fine for people to move from one to the other. The sites are around 100 feet from the parking areas designated for the camping (and none are very near the toilets). Each has or shares a bear box.