If you're renting a car and do not mind driving, you can look into: Plattsburgh International Airport (30min drive), Burlington Vermont Airport (2hrs), Albany Airport (2.5hrs), Watertown International Airport (3.5hrs), Syracuse International Airport (4hrs), Rochester Airport (5hrs).
Yes! Lake Placid and Plattsburgh are both around 30min away.
Lake Placid is full of great places to stay with awesome views of the lake and Whiteface Mtn. You can also check Airbnb for places in Au Sable or the other nearby towns.