For usual expressions, the correct one is "나는 질문하고 있다. 다른 분들은 답하고 계신다." ("I am asking questions. Others are answering questions.") Nonetheless, some others will say that it is correct to say "나는 질문하고 있다. 다른 분들은 답하고 계시다." This seems to be due to 있다's confusing irregularity.
있다 is a verb and adjective (reference). When 있다 is used as an auxiliary verb, it has qualities of adjectives for some conjugations: '-고 있다' is the correct choice to end a sentence (reference). '-고 있는다' sounds awkward in a lot of sentences (reference).
분 is an honorific, and you used it in the subject; this 분 is incompatible with 있으시다 because 주체 높임법 should be used (reference).
The honorific of 있다 is 계시다. 계시다 is a verb according to the standard dictionary while some other dictionaries say it is sometimes an adjective. According to two recent answers of the National Institute of Korean Language (NIKL), 계시다 should follow verbs' conjugations (references A and B). Therefore, '-고 계신다' is correct. (Nevertheless, an interesting fact is that they used '-고 계시다' in this material (표준 언어 예절(2011) - 경어법) and even for this 2018 answer.)