While the photo is handed to Doug's mother it is of two people, not what's suggested.
When Doug is taking his mother to the airport they pass the same antique store not once, but twice.
The priest refers to Doug by name in the confessional. Confession is anonymous.
At the very end, Mr. Kaplan is confronted by Arthur in the same locker room that previously shows Doug, Deacon and Spencer in. According to other episodes, Mr Kaplan is a multi millionaire, therefore he most likely wouldn't be at a mediocre gym/steam room facility vs. an exclusive private NYC club.
Arthur yanks off Spence's towel after he comes out of the sauna. He's standing behind a cart but for a brief moment it can be seen that he was wearing shorts underneath the towel.
When Doug is driving his mother back to the airport, they pass several palm trees and there is a mountain in the background, neither of which can be found in Queens, New York.
When Spence, Doug and Deacon are in the locker room, Spence complains that no one told him that they had left him alone in the steam room, so he was talking to himself "for half an hour." If he didn't know they left, how would he know how long he had been talking to himself?