Who decides what answers and puzzles go in game shows?

I would imagine from the writers which then gets approved by the producers. And sometimes it is the producers who could ask writers to go with certain topics.

I’ve heard once, that in the Netherlands anyway, there are companies that do just that; making questions and such, for all kinds of quizzes and shows.
You go there and buy questions, and I assume after the first purchase they will tailor make them for you.

There are contractors that do exactly this – finding and vetting good questions. In the end, though, the producers of the show make the final decision on categories (Jeopardy) and puzzle words on WOF and such.

I remember reading that “who wants to be a millionaire” had multiple levels of fact checking because the dollar amounts were so high.

I read that Alex Trebek read over every question that was chosen for Jeopardy. I believe there were staff writers who came up with the questions.