Previously I don't think there was a way to do this unless you wrote some java script that you ran from Tasker, but I could never get this to work. (If anyone knows how to write this type of script, I'd love to know.)
However, the latest version of the 'Secure Settings' plugin (by intangibleObject) now supports enabling and disabling multi-window mode (without needing root - if that's a concern). I've tested it on my S4 and it works great. It should work with the S3 as well. Note that this version of Secure Settings is the current beta version which was released only a few days ago. The latest released version on the Play Store doesn't have this option yet.
Also, in answer to the question about listing system settings... you can do this with another app/plugin called 'System Settings for Tasker' by David P. White. In order to actually get the listing of settings through, you'll need to have root access. Without root, you have to know the setting names another way. This app will only monitor setting values. It doesn't have any mechanism for changing those values.
The setting names for multi-window are:
A quick side-note here... if you want to show/hide the little thumb selector (the flashbar) you can long press the back key. This is of course a manual action instead of an automatic Tasker action, but does come in handy if that flashbar is in your way and you don't have a Tasker task set up to deal with it. The Secure Settings plugin action only manipulates the multi_window_enabled setting and not the others.