Fawns will usually keep their spots until they are weaned, and i'm not positive at what age that actually is, but i'm guessing around late August to september sometime, but that's just a guess. Their white spots break up the young's reddish/brown coat. You think of when they're born..usually around May and June, that's about the time when you start seeing new flowers, grasses, and shrubs appear. The white spots help the fawns conceal themselves when the mom is off feeding.
As for the "mange" the deer are slowly shedding their summer coats as fall slowly approaches. You will see them change from the reddish/brown color to their fall and winter coats of a darker gray color.