The second question is whether or not there are Hunger Games, which the novel doesn't elaborate on, but my judgment is that there are not. The victors voted to keep the Hunger Games against the Capitol children when Cohen was in power, and Katniss caused chaos by shooting Cohen with an arrow, an act that had a big impact on the announcement of such events. And it was that Limit Race game designer, a Kepitian, who came into office afterward, and it was thought that he wouldn't have done that to those Kepitian children. And the description of that peaceful life in the postscript would reflect the fact that the game doesn't exist anymore, and has even made its way into history textbooks.
The third problem is that what the novel is trying to say is only clear to the author, and readers will vary from person to person. I think there is telling us to forget the hate, but that's not the only thing.
Gale's words, this I don't really understand why he went to the second district. It could be that he had a good job after the unification of the whole country, and staying in District 12 would have had an effect on Katniss, not only by constantly reminding her of those unpleasant memories, but also by making her embarrassed with Peeta.
Last question, President Snow has been captured and is certain to die, but in a different way. Cohen asked her to shoot Snow, partly as part of her pact with Katniss at the time, and more symbolically in front of the people of the districts.Katniss's earlier conversation with Snow showed her that Cohen was no good, and might even be more authoritarian. She herself on the other hand remains still a pawn. And Cohen's hatred for killing her sister made her choose to kill Cohen.