At the moment, I live by myself, so I just have a bucket that I carry around the flat that I wee/poo into, and a carrier bag for used toilet paper. Then, at the end of every month, I'll empty it out of my window onto the street. I live on the top floor, so this is most convenient for me, but I guess most people have to empty their buckets somewhere else?
Anyway, I realise that most people don't live by themselves, so I'll answer as though I shared with someone else.
In someone else's house I'll usually leave it how I found it, but if it's my own toilet, I'll leave it in its natural, unmolested state after I'm done. I see no good reason to touch/move/alter the toilet seat or lid in any way after I've finished weeing/pooing, so I'll just get up (if necessary), flush, wash hands and leave.
If there is a high risk of splashback from the flusher, or if the consequence of splashback is particularly high in this instance, then I'll close the lid before flushing. Otherwise I'll just flush.