During the limo ride out the side window you can clearly see casino landmarks on opposite sides of the strip. On the gangster side you see Bellagio and then Harrah's.
Charlie Chaplin was (and is) buried in Switzerland, not Las Vegas.
There are no movie theaters in Downtown Las Vegas that would show Charlie Chaplin films. The only theater located in Downtown Vegas is the Eclipse theater, which shows modern mainstream releases, not art house/silent/classic films.
The diner seen in the film is the Omelet House 50s Diner, located on East Desert Inn. It is several miles away from Downtown Las Vegas, although the film tries to imply that it's downtown, with downtown landmarks like the Golden Gate and the Four Queens immediately outside. In actuality there are no major buildings outside of shopping centers and the like in its immediate vicinity.
Many of the aerial establishing shots of Downtown Las Vegas are actually of The Strip, which is farther south.