The correct answer is C. You are requested to shut the door and go to sleep.
The other options are incorrect because they do not accurately reflect the meaning of the original sentence.
Option A, “The door is to be shut and you are to go to sleep,” is grammatically correct, but it is not as concise as option C. It also does not make it clear who is being asked to shut the door.
Option B, “Let the door to be shut and you be asleep,” is grammatically incorrect. The verb “to be” should not be used twice in the same sentence.
Option D, “The door is to be shut and you are requested to sleep,” is grammatically correct, but it is not as clear as option C. It does not make it clear that the speaker is requesting that the listener shut the door before going to sleep.
In conclusion, the correct answer is C. You are requested to shut the door and go to sleep.