Neil is dyingIf you pay close attention to Neil's sprite at the end when the red flash happens, it looks like he takes painkilers... For what? Also, if you pay close attention to other little moments in the game, like when after Eva moves River from Johnny's memory and Neil calls her a poser for visualizing a door after telling him she still had something to do, she wants to know why he's being so melodramatic and fighting so hard... and Neil doesn't respond... It hints that he's hiding something. Of course he could just be addicted to painkillers...
The reason why Johnny is called Joey.As stated by the game, Johnny and Joey were identical twins. Since Joey died in an accident and Johnny's memories were repressed intentionally, it's possible that their mother purposely called Johnny by both names to avoid him asking questions if people called him by the wrong twin's name. The story that the nickname was after his grandfather was likely a cover story in regards to this.
Eva is dying, and Neil is trying to reconstruct her memory, but not according to her wishes.In the real world, Neil loved Eva, but she never returned his affections. So, when she decides to have Sigmund Corp. fulfill her wish, Neil takes the opportunity to relive some of their previous cases and alter things slightly to cast himself in a better light than Eva originally perceived. THAT is why Neil seemed to be so invested in Johnny's fate—to make himself seem like a more caring and sensitive person to Eva. Him sneaking off towards the end of the game had something to do with him managing the equipment hooked up to Eva. As for the stinger, the red flash was a sign of Eva's deteriorating condition, and the painkillers? Well, Neil's been working on the memories for several days now, and he needs to take something to be able to keep at it.