Flynn claims that there's only one Librarian at a time and the next one doesn't take over until the previous one dies, however Judson was a librarian himself and he was still alive when both Flynn and his predecessor were recruited, which suggests that a librarian can simply choose to retire from active field duty.
When asked how Flynn got his position as the librarian and what happened to the librarian that came before him, he states that he got the position because the previous librarian died. While the previous librarian did eventually die, it wasn't until Flynn had already gotten the gig, because he had actually faked his death before.
The Land Rover used in the Black Forest has a license plate starting with JP. There is no city/county in Germany which uses JP.
At the beginning when Baird and her US & German team were going in the building, the flag on the German's SWAT costume was not Germany's flag it was Belgium's.
In a flashback we are shown the librarian that called Flynn was third in line for an interview at the library on the day Flynn was chosen. But in the first movie in which the flashback takes place, the person that is third in line is a short Asian man wearing glasses and a blue shirt not the person who is shown in the flashback in this episode.
It is impossible to tell with certainty if a red dye is carmine just from looking at it. The hue tells you very little since the paint could have been mixed with other pigments. You need a chemical analysis of the very substance to be certain, like with a spectrometer or chromatography.
The outside shots of the museum are from the Deutsches Museum in Munich. However, that is a museum of science and technology where they don't really have paintings. Also it was founded a lot later than 1546. The only 16th century 'museums' in Munich, if you want to call them that, are the numismatic collection and the treasury of the Bavarian Residence. This painting would most probably be found in the Pinakothek, founded in the 19th century.
The exit door in the German museum has a sign which basically says "Smoking prohibited". Even in Germany you are still allowed to smoke outside.
When Flynn is giving the coordinates of the Henge in the Black Forest, he says they are at 84.3 degrees north, 18.5 degrees east. 84.3 degrees north is actually in the Arctic Ocean, very close to the north pole.
In the museum, Lamia directs her henchmen to split up and find the painting. You don't need a SWAT team and a search operation to find a painting in a museum, they have catalogs and you can even simply ask the staff where it is hung.
The Black Forest is some 250 miles from Munich. The Serpent Brotherhood should have beaten the Librarians by several hours in their helicopter.
When Baird got her white envelope with the invitation to work at the library, the invitation said it was to interview for the position, however when she arrived, she was simply given the job without having to do any interview, despite Flynn's protests.