Indy's troop dismounts at Double Arch at what is (now) Arches National Park in Utah. The Balanced Rock is seen, but the smaller balanced rock ("The Chip of the Old Block"), which stood beside it until 1975, is gone.
Indy takes the Cross of Coronado from a cave near the Arch, and flees to a nearby rail line. In 1912, there were no rail lines by the Arch. The closest was 40 miles north. The line there today was built in the 1950s to serve a uranium processing mill in Moab.
Indy's uniform has the Life Scout rank patch. The Boy Scouts of America was founded in the US in 1910, and while it's not impossible for Indy to already be a Life Scout, it's incredibly unlikely.
Panama Hat is driven in a 1915 Saxon Model 14, while his men chase the train in a 1919 Ford Model TT truck. Indy's friend arrives with the sheriff in a 1914 Ford Model T Touring convertible.
Rest of the movie:
The blue stamp on Indy's father's envelope commemorates Texas' 100th anniversary of statehood in 1945.
The Nazis use MP 40s and P38s, as with Raiders of the Lost Ark. The MP 40 was not adopted until 1940, and even though the MP 38 and P38 were around, neither was common in frontline troop in 1938. The Luger was still more common, as was the Bergmann MP35/I, which is used when the Brotherhood attacks the convoy.
The biplane Indy and his father fly from the zeppelin is a Tiger Moth RAF trainer. It's unlikely the Lufftwaffe would have a British biplane as, although there were trade agreements, they had a decent selection of their own planes to choose from.
For that matter, zeppelins only departed from Frankfurt and Friedrichshafen, not Berlin, and, with the exception of the Graf Zeppelin's two voyages in 1929, only made trips to the US and South America, not the Middle East and Africa. Further, its designation is D-138. All zeppelins designations were "D-LZ (number)", and even if this is meant to be consistent with real zeppelins, no zeppelins were built past the D-LZ130 Graf Zeppelin II, and commercial passenger zeppelin travel ceased after the Hindenburg disaster.
The fighter plane that chases Indy in the biplane is a postwar Swiss Pilatus P-2 trainer.
The Nazi convoy has two Volkswagen Kübelwagens (one real, the other a replica built on a Citroën 2CV chassis that gets destroyed). The Kübelwagen was only in the prototype stage of development in 1938.
The book burnings in Bebelplatz took place in 1933. Any books outlawed by the Nazis would have been long since confiscated and destroyed by 1938.
The truck that takes Marcus Brody away has the Afrika Korps symbol (a swastika over a palm tree) on the rear gate. The Afrika Korps was not formed until 1941.
The uniformed woman in the map room of the castle shouts orders to the others. The SS did not have women in any position of power in 1938.
One of the newspaper clipping in Henry Jones Sr.'s book details marriage rates between 1939 and 1943.
The car Indy and his father steal is a 1954 Citroën 11 Légère Cabriolet.
Elsa claims she won the silver medal for Women's 50 Metre Freestyle at the 1932 Olympics for the Austrian swim team. That event was not part of the Olympics until the 1988 Seoul Olypmics.
There were a total of 11 swimming events in the 1932 Summer Olympics, 6 for men (100 meters freestyle, 400 meters freestyle, 1500 meters freestyle, 100 meters backstroke, 200 meters breaststroke, and 4×200 meters freestyle relay) and 5 for women (100 meters freestyle, 400 meters freestyle, 100 meters backstroke, 200 meters breaststroke, and 4×200 meters freestyle relay). Austria did not actually participate in any of the swimming events.
A total of 20 countries competed in the swimming events. The participants of the swimming events were Japan (12 metals), United States (10 metals), Australia (2 metals), the Netherlands (2 metals), the United Kingdom (2 metals), Denmark (1 metal), France (1 metal), Hungary (1 metal), the Philippines (1 metal), South Africa (1 metal), Argentina (0 metals), Brazil (0 metals), Canada (0 metals), Finland (0 metals), Germany (0 metals), India (0 metals), Italy (0 metals), Mexico (0 metals), Norway (0 metals), and Sweden (0 metals).
The bike Indy steals is a 1988 Dnepr MT-11, which, as detailed on the Raiders of the Lost Ark page, is a Soviet continuation of the IMZ M-72, a copy of the BMW R71, which was rejected by the Wehrmacht in 1939. The sidecar in particular is very un-BMW. The bikes used to chase them are 1977 Honda XL 500S trail bikes.
The car Indy drags the knocked out SS officer behind at the Berlin book burning, after stealing his uniform, is a 1949 Opel Kapitän.
At the airport in Berlin, two people are reading newspapers dated 1918.