Hat box?
A hat "drop" box. I know this sounds crazy, but just the bottom part of a hat box will preserve the hat and you won't encounter the I-have-no-time-or-patience-to-put-it-away-neatly mentality. If putting it upside down still triggers that mentality, you can remove the hat insert and just place (throw) it in the empty bottom half of the box.
Coat: Try keeping a hanger on the door (ceiling) hook at all times. You won't need to search for the hanger or spend the time trying to fit the hanger curved part over the door hook each time.