To what extent is working on a quiz show permissible, given that it includes questions on arts (music and visual arts), history, religion, geography, science, general knowledge, and literature?

There is no doubt that means share the rulings of their ends, and whatever leads to the forbidden is likewise forbidden. No action that aids in the forbidden or cooperates in sin is permissible. Since your work involves entering questions and answers that include the identification and praise of immoral people, such as male and female singers and directors of indecent works, this work involves cooperation in sin and falsehood. If you can avoid this prohibited act, then do so. Otherwise, we advise you to leave this job and seek another one free from such matters.