The correct option is
C whose
As we can see, the answer will connect the first clause to the noun 'house'. This can only be achieved by using relative pronouns and thus, the answer must be a relative pronoun. Further, the sentence indicates that the house belongs to the subject's friend and so, the answer must indicate possession.
Option A is incorrect. In 'who lives in', 'who' is a personal pronoun of the nominative case. Thus, it does not indicate that the house belongs to the subject's friend. Thus, it is incorrect.
Option B is correct. 'whose' is a relative pronoun that indicates possession. Thus, it meets the required conditions and is the correct answer.
Option C is incorrect. 'who is' has a personal pronoun in the nominative case. Thus, it does not indicate that the house belongs to the subject's friend. Further, 'who is house' indicates that the friend is the house. This makes no sense at all. For these reasons, it is incorrect.
Option D is incorrect. In, 'in this', 'this' is a demonstrative pronoun and not a relative pronoun. It is used to point at objects and does not indicate possession. Thus, it is incorrect.