this English sentence is clumsy and not really very good English

Are you a teacher by profession? = this would also be ok in English.

Is the article really needed here?

I agree with everyone below who says that 'Is your profession a teacher?' is not a good translation for Ваша професія - викладач?

The English sentence 'Is your profession a teacher?' COULD have a meaning for a native English speaker, but not that meaning... One possible meaning of it might be: "Does your profession [which could be e.g. working as a bus-driver or as a surgeon] teach you anything? [i.e. "Does your experience as a bus-driver or surgeon teach how to live your life better, e.g. be more compassionate etc.?]

In fact I believe that , as a translation for Ваша професія - викладач? 'Is your profession teacher' - without the indefinite article - is more accurate (if a little clumsy)

Is teaching your profession?,

I'm not a native speaker (of English or Ukrainian), but this sounds like very awkward phrasing to me. Is this how people say it in Ukrainian?

Why not translate it simply as:

"Are you a teacher?"

And so on for the other professions :-)

Well, apparently Doulingo is not always a teacher by profession... "Ваша професія - викладач?" translates to "your profession is a teacher?" based on all the lessons I've learned so far.