For sleep: Cooler temps can help you lose weight (the body burns more energy to keep your core temp at a cozy 98.6 degrees), can help you sleep better and may even improve your metabolism while you sleep.
For studying or working: Our brains work better in cooler temperatures, too. Science says 62 degrees is the magic number for brainwork. A study found 62 to be the temperature where people functioned optimally.
For skin: Most of us think that cold weather leads to dry skin and wrinkling, but a Harvard Health Letter from 2010 includes the following interesting info: "... moderately cold temperatures could be good for the vasculature because it trains blood vessels in the skin to be responsive."