Katara was able to learn the technique so quickly, only knowing of its basis; however, remember that Katara is a proficient healer and therefore has an advanced knowledge of the human body, along with the fact that she has advanced expeditiously in her studies - she mastered it at a first try, even being able to bloodbend Hama at ease. This may be possible because of the Moon spirit, Katara bended water from the oasis and touched it which could be like a powerboost to a waterbender. Katara learned it faster than Hama who needed alot of practice before she could perform it on a Fire Nation Gaurd, And for Katara to perform it on a master Waterbender on a first try could probably be from the special water. Katara could have done so in the day if she practised more. Katara was also a proficient healer so she had to know blood flow and stuff like that.
Yakone could have been the same, or if he used the chakra gates to stengthen the power and flow of the benders chi. Chi/chakra would affect more than just the power of a bender. It doesn't just effect the Avatar , but can effect benders and even non benders. Amon may have resisted this for the same reason, the only difference being Amon is a non-bender and instead of powering his bending he power the way his body works.
I have a theory that the reason Aang waterbended without the moon was because the avatars released the seven chakras whilst he was in the avatar state. My point being is that the moon stands as a temporary link to waterbending and once you master all chakras, you wont need the moon.