I tend to put both the seat and lid down before flushing because someone told me once that when you flush, tiny particles spray everywhere... So, just to be on the safe side.

I think the main reason women insist that the man puts the lid back down is that no one wants to touch the toilet seat, haha. I never really thought about it, but I only share my bathroom with my little brother and I think he's taken to closing the lid following my example.