"I am a sleeping man."
Translation:저는 자는 남자입니다.
Yes, just like using multiple -ing words, for both adjectives and verbs, is acceptable in English: The waitING man is readING an interestING book. (기다리는 남자는 재미있는 책을 읽고 있어요.)
While the answer to your question is yes, the second 는 in the given sentence is not a particle.
The second 는 is not a topic particle. It is the Korean method of changing a verb into an adjective, of changing an action verb into a descriptive word. In answer to your question, it is indeed possible, maybe even common, to have more than one 는 as a particle in the same sentence.
This sentence is technically correct, but wouldn't it be wonderful if the course developers had chosen sentences that are actually used and useful in both languages. If 1000 Koreans and Americns were canvassed, I doubt you would find even 10 who had ever uttered either phrase.
The sleeping man worked all night. The man sleeping is sick. The sleeping person is a man. There are so many possibilities. --or change the verb. I am a working man. I am a joking man. I am a welcoming man. I am a drinking man.
All sentences are useful as long as you pay attention to the patterns. Plus, this sentence helps learners differentiate between sleeping (자는) and I (저는), which is incredibly useful. Moreover, you can use this idea to describe yourself in a picture: 사진에서 저는 자는 여자입니다.
Of course. It would be nice if there were clues about the level of formality desired.