"She takes notes quickly."
はやく、means both quickly and early. But here in this context it talks about 'speed' therefore you cannot say 'early'. すばやく、はやく、means almost the same but すばやく is, like you say, quickly. In very fast speed.
ノートをはやくとる、とるのがはやい、are quite similar to me. It is something like the difference between He plays tennis well. and He is a good tennis player.
Could someone explain the structure of the sentence please? I did not understand why のが is needed here, wouldn't it turn "takes" into "taking"? The sentence should then be : "As for her, taking notes quickly?" I know literal translation doesn't always work but "takes" should simply be とります, and you could say for this sentence : "かのじょはノートをはやくとります". Google translate confirmed this, but not sure if to trust it.
Usually “は” indicates the major subject, while "が" indicated the minor subject, like the subject in a clause in English. So this sentence literally means, like you said, "As for her, taking notes is quickly." And "の" transforms "ノートをとる" into a noun phrase so that it can work as the minor subject, just as turn "takes" into "taking" in English.
In my opinion, "かのじょはノートをはやくとります" means slightly different from "かのじょはノートをとるのがはやいです"。The former only describes the action that she's taking, while the latter indicates that taking notes quickly is one of her characters. I'm not quite sure about this one