Lister, you can fly the cub all day long with both the window and the door(s) open. I recommend warm weather, though.
The right side upper door stays locked up if you're flying with the door open, but you can also close the upper door and leave the lower open, or close the lower and leave the upper door open, instead.
A handy feature of the doors is that the lower door serves as an ad hoc stall warning device. If you haven't tried it, leave the doors open, and practice stalls, you'll find that the lower door starts to float as you approach a high angle of attack, then buffet, and then drops as you stall.
There's nothing better than flying low over alfalfa fields at harvest time in the morning with the cub doors and window open.
So far as "prop wash," fly with the doors and window open, and put your hand outside the cockpit while you're flying. Pull the power to idle, and see if you can detect the change in airflow. Most of your airflow into the cockpit is a combination of propeller blast and slipstream. You'll note a big difference, however, between climb or cruise power,and an idle power setting.