The correct option is c. 's sent / has he been looking.
Explanation:
In the first part of the sentence, "He's sent hundreds of job applications," 's sent is the correct form (He has sent), as it refers to a completed action with relevance to the present (he still has no answers). In the second part, "How long has he been looking for a job?" has he been looking is used to ask about a continuous action that started in the past and is still ongoing (he is still looking for a job). Thus, option c is the most fitting and grammatically correct choice.