U.S. enlisted service records of the Second World War were maintained as pocket sized booklets which could be easily stored and carried. When Smith holds Hawthorne Abendsen's service record, the folder shown is actually a type of U.S. service record used from the 1960s through the 80s. This would not make sense since the US. military has ceased to exist after World War II.
The "real"-timeline Dornier Do 31 was not built until 1968. However it's clear throughout the series that aviation in the GNR is more advanced, as can be seen by the use of supersonic passenger planes. It's thus possible that the DO 31 was built as early as 1962 in the alt-timeline.
Dr. Mengele asks the officers to hand over their weapons before activating the Machine, because the electromagnetic field that would be created would pull all metallic objects towards it. However, none of the medals on John Smith or Himmler's uniforms are affected. It should also be noted that there is a total of about 4-5 grams of iron in a healthy human body as well (mostly in red blood cells and muscle cells), which should also have been affected by the electromagnetic field: essentially, it would all be sucked out of the body, killing everyone in the room. Magneto did something similar in X2 (2003).
Since an integral rule of travel is that an individual can only pass into another world where their twin has died or never existed at all, Trade Minister Tagomi should not have been able to travel to the universe where his family (and, most importantly, *himself*) are all alive.
However, Tagomi himself has stated that both his wife and son died years ago and our-timeline Juliana explains to Tagomi that when he left she lost him on a bridge and feared he had jumped. It's safe to assume that our-timeline Tagomi did indeed jump and therefore die.