Do I need an egress window?
Well first of all for safety reasons you seem covered.   You have exits in two directions and possibly three if the regular windows can fit a human.
From a code aspect there is nothing that says you cannot put a bed in your basement.   Now there are real estate listing rules that do not count a living space as a bedroom in a basement unless it meets certain criteria.
For most home listings there must be a door that you can walk out of in the basement for a room to be considered a bedroom in a basement.   You seem to have that in the door to the garage.   Also allowable is an egress window.
That being said you can put a bed where ever you want and there are no laws against that.