2&3 are both "here's what the others were doing while War was chained up for a hundred years". Their stories are set in motion because of the premature apocalypse, and will probably have implications for humanity further in the future, but there's no impact on War directly, aside from 3 implying
spoiler
how Strife lost his guns for you to get them in the first two games
As for Genesis, the main thing you should keep in mind is that it isn't a mainline game. It's more an isometric twin stick shooter/hack'n'slash, and the gameplay isn't as deep as with the main games. But it's got co-op, if you have someone to play with, and it does provide some characterization for Strife as well as showing an example of the sorts of missions the Horsemen took for the Charred Council.
Damn, really? That sucks, thanks for the heads up.