The scenery and pavement visible through the car windows as Leonard, Sheldon and Amy travel back to Pasadena shifts from photographic to CGI generated scenes. The scenery shifts in between words and sentences of dialog, far faster than any actual vehicle could travel.
The theft of Sheldon's items happened in the train. Amtrak police would have investigated the theft, not the local police.
There is no hot dog stand in the Salt Lake City train station. If there were, it would not be open when Sheldon arrived. The west bound Amtrak train (California Zephyr) is scheduled to arrive at 11:05 PM and the east bound Amtrak train (California Zephyr) is scheduled to depart at 3:30 AM.
The Highway scenery on the trip back from Kingman AZ shows Leonard driving down the center of the two lanes.
All four headrests are missing from Howard's car.
There is no passenger train station in Des Moines and no passenger train passes within 45 miles of Des Moines. If Sheldon never left the train station, he would not have been anywhere near a fire in Des Moines.
Sheldon becomes stranded in Kingman, Arizona which is on the western side of the state bordering California. On the ride home to Pasadena, the background scenery (after driving two hours) consists of reddish mountains. Assuming they are taking Interstate 40, this is not what one would see west of Kingman. However, it is similar to what one would find hours away in the opposite direction on Interstate 40 when approaching the New Mexico border.
The only direct route from Kingman, Arizona to Southern California requires the use of Interstate 40 and Interstate 15, yet Leonard, Amy, and Sheldon are driving back to L.A. on what is clearly marked as an undivided four lane highway in a rural part of either state.
The highway scenes after Leonard and Amy pick up Sheldon in Kingman are actually US-191 between approximately 6-10 miles north of Moab UT. Also, the first scene with all three characters in the car is shot 10 miles north of Moab and the 2nd scene is shot only 6 miles north of Moab and they are in the northbound lane of the highway.
In order for Sheldon to be robbed of the few possessions he started on his trip with, in an Amtrak sleeper he would have had to either have left them outside his room or left his compartment door unlocked, both of which seem unlikely for someone as obsessed with detail as Sheldon.