The signature that Erich Kästner gave a woman on an autograph picture is different that Kästner's actual signature. However in the signature Kästner was writing into Hans' book is the same as the one of the real Kästner.
In the book store where Kästner tells the clerk about his "new boy Emil" there is a copy of "Das öffentliche Leben" (The public life) by Heinrich Mann in the window. This book was released in 1932 while "Emil and the Detectives" was released in 1929. In the next scene the movie adaptation of "Emil" was mentioned which would set the scene in 1929-30, two years before Heinrich Mann's book was released.
Erich Kästner mentioned to Hans that he is writing a funny screenplay for Heinz Rühmann, which would be a direct reference to Ich vertraue Dir meine Frau an (1943), where Heinz Rühmann is playing the lead. The scene is set in 1936 but the movie was released in 1943. However he could have meant that the writing process is going on, before the movie is released. The movie had several screenplay versions.
When Hans is reading Kästner's poem "Die andere Möglichkeit" (The other possibility) to his class mates and in Kästner's flat is is missing out a few verses. In his flat he was only reading the first and the fourth and in school only the fifth and the last. There were no cuts between the two scenes and verses.