During a radio broadcast (some 4 minutes into the movie), the Martin Sheen character gives the name of East Germany - the country of his birth - as "Deutschland Democratic Republic" whereas the official name was "German Democratic Republic" in English and "Deutsche Demokratische Republik" in German.
Several. First, it is not likely Alex Holbeck would have used a pseudonym like Janosch as it's Hungarian, not German. Second, many of the plots hinges on the GDR just dropping the ball. International trains crossing into the GDR stopped at the border and stopped at least for an hour. Alex's miracle escape in Berlin would not have worked, no passengers could leave the train and the border guards were very thorough. Suppose he jumps that hurdle despite the KGB being on high alert because he returned, no amount of disguise would have helped. Trains didn't stop at a fictional Ostbanhof but Friedrichstrasse where entering the GDR meant a second passport check. The well known female protagonist is a subversive element who either would have never gotten a phone assigned, or it would be wiretapped and a KGB agent would know that. No civilian vehicle could ever get near a military airport, those were exclusion zones. Last, but not least, the Stasi was very proactive, them not knowing about an impromptu Christmas party at the embassy is impossible. In real life, they have infiltrated people peacefully pretending to be Native Americans because they believed it was a CIA subversion.
One of the aircraft seen on the tarmac claimed to be a "Mig" is a French Dassault Super-Mystère B2 with the unit badge of Escadron de chasse 3/12 Cornouaille visible. This is because the scene was shot at Le Bourget in France.
When Martin Sheen arrives in Berlin, the luggage he is carrying has printed on the side "University of Michighan" which is a misspelling of "Michigan".