So the movie never elaborates on how Goodkat knows Lindsey took a picture of him in the elevator. Then it hit me. Phones don't work in there. —Truan
Yes it does. Goodkat says "I had that same phone," (or something of the like) indicating he knew the functions and tricks.
After Goodkat shoots Lindsey, it seems like Fridge Logic that he doesn't bother to check whether his victim is truly dead. But here's where the Fridge Brilliance kicks in: most likely Goodkat does see that Lindsey is still alive, but seeing this makes him realize that Slevin had warned her about the hit. At that moment Goodkat knows that Slevin must really love Lindsey for him to disobey his mentor like this, and that's what makes him spare Lindsey: he can relate to Slevin's feelings for Lindsey, because similar empathy made him spare Slevin 20 years ago. So after processing all this in his head, Goodkat slips away from the morgue, letting Lindsey think she has fooled him. This also explains why, after Slevin asks Goodkat how he found about Slevin's plan to spare Lindsey, Goodkat just says: "I'm a world class assassin". A world class assassin wouldn't just shoot someone once and leave, without checking out the victim really is dead.
Though it takes a second watching to pick up, there are 3 very subtle clues as to Slevin's identity; 1. Henry is seen playing with a baseball mitt and is promised a trip to a baseball game; 2. A bookie is killed by a very fast, very well aimed baseball, suggesting the killer is probably a big fan of the sport; 3. When Slevin is told to play ball, in a quick and literal-minded fashion, he quips "You think I'm tall enough?", suggesting he has a ball sport on his mind.