Johnny Depp signed on without reading a script. He wanted to be a part of the series because he is a self-proclaimed massive fan of the series.
Tom Riddle (Lord Voldemort) is estimated to be less than a year old when this movie takes place. In Harry Potter and the Chamber of Secrets (2002), which takes place in 1992, the Chamber is said to have been opened 50 years prior, i.e. in 1942. It is also said that Tom Riddle turned 16 that same year, so he must have been born in 1926. Since this movie is set in 1927, Tom would be one year old by the end of that year.
When Leta opens the desk in Hogwarts to view the engravings, a symbol similar to the mark of the Deathly Hallows can be seen as well.
Newt Scamander carries a wand, practices magic, and works for the Ministry of Magic, even though he was expelled from Hogwarts, which puzzled several fans online, especially since Hagrid wasn't afforded those liberties. J.K. Rowling explained that the reason for this will be revealed throughout the series.
To prepare for the role of Dumbledore, J.K. Rowling gave Jude Law extremely secret details about his character during a two and a half-hour meeting between the two of them, during which he was allowed to ask as many questions as he needed. Law also watched Michael Gambon's performance in the previous four films to help establish the character and performance. This meant he added a slight Irish inflection to his dialogue, just as Gambon had done to maintain continuity, and pay tribute to Richard Harris, his own predecessor in the role.