Nobody in their right mind has the time and space to properly sort and clean the trash. For the rest, inefficient use of water and detergent has an environmental cost too.
This stuff should be all handled by a combination of centralized work at sorting plants and alterations to packaging. Containers can be cleaned more efficiently in a centralized location, and as for the pizza boxes, perhaps it should be mandated that pizza boxes must be fully lined with aluminum foil on the inside - this would be a win for both recycling and product quality, as the pizza would stay warm for longer.