Antecedent of "which" in "A man coming out of the backdoor of his house which is facing east"

That is too ambiguous. It is impossible to figure out if it is the house that is facing east or the backdoor, unless more details regarding of the direction are given.

A man coming out of the backdoor of his house which is facing east witnessed the beautiful sunrise.

[updates]

Given that this question appeared in an exam and it is about direction, the most logical way to analyze it is to perceive it from the perspective of the question setter. If I were the question setter and I was planning to have choice B as the correct answer, it would be more sensible to set the question as "A man coming out of his house which is facing east...", instead of giving extra information about where the man came out of his house (which can have an impact on the question as these very few words can change the answer from B into A) and later assigning east to the direction the backdoor faces, making its answer the same as of the question that is not as explicit. Therefore, it would be more logical for the question setter to specify the direction which the house is facing but not the backdoor's. And thus I would go with option A.

Regarding of the second question: I don't think a comma would make much difference in this scenario.

It is clearly the door which is facing east, which gives enough information for the calculation. Otherwise, a back door on a house could face any direction. Just because the door is at the back of the house doesn't necessitate facing directly opposite to front of house, even if it was a simplistic four sided house.