Some of my more strapped households prefer to save their leftovers for later on, even if it's been partially eaten. However, when telling them to clean up multiple dishes, they often will grab the half-full plates along with the empty ones! Is there a mod to stop this?

Also, is there a way to stop them from dropping plates of food on the floor when there's a counter available in the next room?

Not that I know of. Half eaten plates count as dirty.