Short answer: You needn’t worry about “needn’t”. “Don’t need to” and “needn’t” are both present tense, and have the same meaning, but “needn’t” is less common (especially in America). “Didn’t need to” and “needn’t have” both refer to the past. “Needn’t have” can be used to clarify that the action was in fact performed unnecessarily. “Didn’t need to” is ambiguous as to whether or not the action was in fact performed. In the US, we rarely say “needn’t have”. In fact, “needn’t” and “needn’t have” are even going out of style in British English these days.

I agree that “needn’t” constructions are probably more common in British English. But other than that, Needn’t = Don’t need to Needn’t have = Didn’t need to

The only difference is the tense, eg. I didn't need to pay an entrance fee (past) You needn't pay an entrance fee (current) They both give us the same information - it was free/ it is free In both, a lack of obligation is expressed:)