I think they are all valid, depending on context.
Different types of questions
"There are many different types of of questions in the examinations." This relates to the plurality of both words, there are many types of question in many examinations.
Different type of questions
(I cannot think of an example for this version. All my attempts worked better without the "s" on "questions".)
Different types of question
"The examination has many different types of question." This relates to the plurality of "examination", there is one examination in this case, but it has many questions.
Different type of question.
"To understand xxx requires a different type of question [than the one you are asking]." or "... [than those that you are asking].