"Die Frau lernt."
Translation:The woman is learning.
You don't know from the words itself, because learning is a form of time not existing in german. You can say both and depending on the situation both translations are correct. ..ing if sb. is doing it just now, so if the woman sits in front of her book, she is learning. Maybe she just learns that you don't walk over red lights by gettING hit by a bicycle. In german one would circumvent this by adding "just now" or "at the moment" or sth.
The woman teaches . I think it's an appropriate verb used in this case. What do you think so?
The verb lernen means to learn, not to teach which is lehren. It can mean "the woman learns" however.
Why is "The wife learns" incorrect? Is there a difference between "die Frau" as "the woman" and "die Frau" as "the wife"?
This is because the verb is in the present tense. Lernt may sound like the English "learned" but keep in mind that doesn't mean it has the same meaning as English. German and English are two different languages.