Men, women, and everyone in between and between all sit at some point when using the toilet.
The most efficient thing to do is leave the toilet seat the way it is when you’re done. Minimal effort.
The most fair thing to do, oddly, is to leave the seat in the opposite position it was when you got there; everybody flips it once, it may be before or after you use it. Fair.
It’s unsanitary to flush with the lid up.
I’m in this camp though, so it’s always left down.