In the case of an object that can reside in two positions or states, the verb "to leave" can indicate an action resulting in the state of the object having changed from one position to the other. If I come upon a basketball lying at center court in a vacant gym, I can "leave" it in the ball rack (its proper place, since no one's playing or practicing). If I attempt to valet park and absently take my keys with me, I can go back to my car and "leave" them in the ignition. If I leave the house at night and I turn off all the lights, and my wife is out as well, and she is expected to be home before me, I can go back in the house and "leave" the lights on for her.
It is in this context that I have used the verb "to leave" with respect to the toilet seat. If you are a male, you place the seat into the "up" position to urinate, and then you "leave" it down when you are finished.
As to the covers, they must be quite thin and not too shaggy or they will indeed fall, imperiling one's manhood, possibly rendering moot any discussion of toilet seat placement.