This seems like something that should be true, but I think I remember seeing a Mythbusters episode where they decided it didn’t make a difference. That show was more about entertainment than science, so I wondered if there was a more rigorous study done? I’ve definitely seen splashes of water(?) come out from flushes so that alone seems to argue for closing lids.
you don’t, most of the time is not needed. if it is, a sencond lighter flush with the brush thing will take care of everything that is still there. with a controlled flush you just let some water run.