If I make a question and answer game based on a mix of popular licences such as Harry Potter, Marvel, Lord Of The Rings and so on, do I need rightsholder permission?
In the United States, you almost certainly do.
The most on-point case is Castle Rock Entertainment Inc. v. Carol Publishing Group, 150 F.3d 132 (2nd Cir. 1998). That was a case dealing with Seinfeld, where a publisher printed and sold The Seinfeld Aptitude Test, a book of Seinfeld trivia. The publisher argued that the questions and answers were simply unprotected facts about the show, not copyrightable expression, but the court concluded that the facts were merely restatements of the expressive works of the show's creators. Although the book was in some senses transformative, the court found that the transformation was "slight to non-existent":
[T]he fact that The SAT so minimally alters Seinfeld's original expression in this case is further evidence of The SAT's lack of transformative purpose. To be sure, the act of testing trivia about a creative work, in question and answer form, involves some creative expression... However, the work as a whole, drawn directly from the Seinfeld episodes without substantial alteration, is far less transformative than other works we have held not to constitute fair use.
A major part of the problem in that case was that the facts suggested that the publisher was not looking to create an educational or critical work revolving around Seinfeld, but rather to provide Seinfeldesque entertainment to Seinfeld fans.
There is some discussion in Castle Rock suggesting that certain behind-the-scenes facts might change the analysis, but from what you're saying, it seems doubtful that you'd have a different outcome.