She thinks it is a waste of time to go back and see the horrible deaths of innocent children.
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If you really want to know you should read the books.. But yes she does go twice once for her sister and then one after the country starts having uprisings because of her she & Peeta go both times and they both survive both times
Katniss and Peeta's main connection is the bread. When they were little, Peeta had a huge crush on her, which was unknown to Katniss, and he took a beating from his mother and gave her a loaf of bread because she was poor and had no food.
Haymitch trains Katniss and Peeta in the first book because he is the only living past winner from District 12. All trainers must be winners from their district, but since there's only one, Katniss and Peeta have to share one.
Actually, no one wins the 75th hunger games because on one thing we're sure of is that Finnick and Katniss were rescued by Haymitch during the Games. And Peeta was definitely captured by the Capitol during the Games.
That the victors who won the Games in the past are going back into the arena. Normally, the deal is that if you win the games, you're out of the arena for good. But President Snow changed that rule in order to get rid of Katniss in a way that wouldn't arouse suspicion, since she was an idol in nearly all of the Capitol