Well I think it depends on the exact circumstances.
At work I prefer to go into a trap with the lid already up. That way I don’t have the worry of what I’ll find when lifting it.
At home I normally put down seat and lid (and close the door). Just seems tidy. But at night when I jump out of bed needing instant relief, it’s better if the lid and seat have been left up.
As for flushing, I do so before closing the lid. I need to check if my fibre-enhanced stools will make it round the bend, rather than leave it for someone else to marvel at. This talk of aerosol spraying of faecal detritus seems a but overplayed – the stuff is under water for a start.