I think people put too much emphasis on Neil's shitty story telling abilities. The dude doesn't have Alzheimer's, he just sucks at telling stories and has forgotten some of the minor details (from 20 years ago).
We get an example of that when he tells the mc about running off with Lynnete:
Neil: "It took her three days to run away from home and track me down."
Neil: "No...wait...five days?"
Neil: "Hm...maybe it was something along the lines of a week?"
mc: "Are you seriously asking me that?"
Neil: "No, well, it was something like that."
Neil: "Either way...where was I?"
Neil: "Oh yeah!"
Neil: "She didn't want her dad's life or his money..."
Neil: "She wanted me."
He's not senile, it's not like he's mumbling in his soup about aliens or some shit...
Also the idea that the mc's birthdays are bittersweet for Neil is because it's both the day that his son was born and the day that his wife died. I thought that was fairly obvious.
Why would Lynette be fucking someone else in the few days between Neil leaving and her giving up everything to be with him?
The mc got into a fight at school and nothing came of it? Big deal. How many people had charges pressed against them for getting into a fight at school? No one I ever met that's for sure.
The diary was probably left behind when she ran off with Neil. The mc's grandparents (Lynette's parents) probably sent it to him when he turned 19 (strange it wasn't 18, but whatever).
The bit with Burke was him emerging from a room after potentially having sex (or failing to have sex) with Mona. I think worrying about the mc and the sprinklers was the last thing on his mind: