Rob McElhenney (Mac) has lost the weight he put on for season 7 by this episode. He previously gained about 50lbs for comic purposes.
Charlie's timeline for when he got the dog painting is perfectly plausible. He claims he got the dog painting in S1E6 "The Gang Finds a Dead Guy" which is believable because it does not show up in his apartment before then. There is also a flashback to S2E8 "The Gang Runs for Office" showing that he had it at that point.
When Mac and Charlie are in the dentist's office discussing the casting in the Nazi movie of Pop-Pop, Charlie (Charlie Day) notes he wouldn't mind playing Hitler. Going the Distance (2010) had Day's character mistakenly showing up at a bar dressed as Hitler when he meant to be Charlie Chaplin.
The clock in the dentist office reads 4:24 and doesn't move from the time.
The Final Solution (German: Endlösung) was a global plan during World War II to systematically exterminate the Jewish population in Nazi-occupied Europe through genocide.