In reality, Finny had always known that there was a war, he was just playing an elaborate denial game with himself, in order to deal with not being accepted into it. Not being accepted into the war is too hard for him to face, so his way of coping was concocting a story about it being fake. However, what makes him kind-of snap out of that denial is hearing about how basic training had pretty much caused Leper to snap and lose it. That was pretty shocking, and Finny states of it,
"When I heard that about Leper, then I knew that the war was real...if a war can drive somebody crazy, then it's real all right."
If Leper, their calm and eccentric friend is pushed over the edge because of the war, then its reality is brutal and undeniable, and it is what forces Finny to admit its reality to Gene. I hope that helped a bit--for your other questions, try submitting them separately, as the format of this website allows for one a day. Good luck!