Around the end of Book 1.
Aang first starts learning waterbending in Book 1: Water, specifically in the episode titled "The Southern Air Temple." Katara teaches him some basic techniques while they are searching for the waterbending scroll.
Pakku. Not Katara.
Due to the fact that Aang is the avatar and waterbending is apart of his genes and Katara is a waterbender the most likely out come would be a water bender child but scientificly it would depend on the dominate and recessive genes from both of them
Katara gets burned in the episode "The Deserter" Aang is trying to learn firebending and accidently burns her, but after that she learns that she has healing abilities by dipping her burned hands into water.
Well zuko never "gave back" her necklace, Aang stole it back in the episode "the waterbending scroll" and gave it back to her at the very end.
yes
Waterbending uses the moves of Tai Chi.
he has a vision of her when he is in the swamp in Book 2 chapter 4, but actually meets her in book 2 chapter 6.
Aang accidentally burns Katara with his firebending in the book three episode "The Firebending Masters" (Season 3, Episode 13) of Avatar: The Last Airbender.
She probably learned while on the set of the movie.
Aang's dream? If you are refering to Nightmares and Daydreams then it is chapter 9, season three.
Korra will be taught air-bending by Aang's son,Tenzin.
On the extras version on the first episode, the Boy in the Iceburg, it claims that it was "love at first sight." But a big moment for her relizing its meant to be is in Book 1, Chapter 14, "the Fourtuneteller". Where a fourtuneteller named Aunt Wu, predicts Katara will marry a very "powerful bender", and later in the episode, Sokka comments on what a "powerful bender" Aang is and Katara is in shock for a second. But, in Book 3, Chapter 16 and 17, "the Southern Raiders," and "Ember Island Players," it is hinted that Katara has gotten over Aang for Zuko, but in Book 3, Chapter 21, "Sozin's Comet, Part 4: Avatar Aang," it hints that Katara has fallen for Aang one again during Zuko's coronation.