Translation:The woman works out.
The emphasis here is on the exercise, not the woman. This sentence answer the question "Who is excersise?" The answers could be a woman, a man or even a dog (which is normal here in Duolingo:). If the answer is "The woman exercises", then the question should be "What is the woman is doing?". This is how I understand the difference between both annswers. Hope it help you and correct me if im wrong :)
No, a typo is not given if you change the conjugation of the verb or create another legitimate word. If you had written “worsk” or “wrks” that would have been a typo for “works”, but “woks” is another legitimate word so that would not be a typo either. The s is required on verbs for third person singular such as any one person that you are talking about.
Both "works out" and "exercises" are correct. However, the mobile app only gives word tiles for "works out".
And currently (2019 November), keyboard input for answering in Korean is only on Duolingo website, not mobile app.
Note also, the English verb in present tense for second person singular (he, she, it) adds the "s": Works out, exercises, talks.
Here are few that make additional changes: I do, she does; I have, she has; I am, she is.
As for articles in English, it can be "A" or "The". I do not see any definite clues in the sentence for one or the other.