Someone is playing saxophone on the recording of 'Blue Moon of Kentucky'. None of Elvis's Sun Records recordings featured any saxophone.
At about 14 mins, when Sam finishes using the diner telephone and is about to look into the mirror to see Elvis's reflection, the actor playing Elvis can be momentarily seen getting into place alongside Sam, in order to be present for the reflection.
Elvis made his first recording with Sun Records in August 1953, not July 1954.
Although the episode is set in 1954, the pinball machine in the diner is a Gottlieb "Baseball" introduced in 1970.