"Not quite yet" vs "not yet"

I believe this sense of "quite" is as used in both UK and US English (see 1st meaning here for definition), meaning "completely", so that "not quite yet" implies that the task is not completely finished, but also that the majority of the work is done. "Not yet" is simply a more polite way of saying "No" in this case.

So to answer your questions 1. Yes, it does mean that it will be done shortly. 2. Yes, it does imply less time than "not yet"

NB: Definition linked above is this:

quite /kwaɪt/ adv to the greatest extent; completely or absolutely: you're quite right, quite the opposite