"Do you know that woman's name?"
Continuous form is always used when describing that you "know" something (because "knowing" is considered a continual state). It is used not in -te form in past tense (知りました or 知った, depending on formality) to describe the idea of learning something for the first time, but in past progressive when describing something that you knew in the past. In present tense, you will usually use present continuous.
Not sure ! It might depend on the fact that, if the person you are looking for her name is in the room, but if you are talking about a person that you are describing (as example you met her in a meeting some days ago), you shouldn't 知って but しり. Context here is not known. Ambiguity in DuoLingo lesson (not the only one...) ???