A verb is used to express an occurrence, action, or state. It is the grammatical centre of the predicate in a sentence.
An auxiliary verb is a verb which gives is used in the formation of tenses, moods, tones, etc.
Option B is correct as smoke is the main verb here. Therefore, using not is necessary and as Bill's father is singular, it can't be followed by do. Also, isn't cannot be used here as the blank is followed by smoke, a verb.
The other options are wrong as his father and Bill's father indicates that we are talking separately about the two and that the verb following shall take s.
The correct answer is B)smokes/doesn't