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Did you check for spark? cables? and plugs?
check the gas cap maybe its loose.
did you check the ecu? are you getting any codes?
sometimes the sensor gets dirty and all it needs is a good cleaning go down to your local auto parts and buy a sensor cleaner try that before replacing the sensor
could be many different things, do you have fuel and spark? i would try to hook up the car to a diagnostic machine that would help you determine what the problem is.
You could buy a reset tool, go to the dealership or to a local shop who has the machine to reset the light for BMW's
I would try to get a diagnostic to see if you have any codes go to your local auto zone they offer free diagnostic, it could be many things the diagnostic will help you determined what the problem is.
it could be the water pump, fan, or air in the system.
Labor depends on your location but here in the Bronx you can get that done for about $800, i would try to get the engine used from a local junk yard google LKQ they offer good parts with warranty
it could be one of several things use the process of elimination, it could be the fuse, the brake light switch, the brake light bulbs or it could be the module for the brake lights if your car has one for that system