I decided to query the local WSDOT office out in Wenatchee, and the director replied back with the information I needed. For anyone interested in doing winter driving through the mountains, the link is below.
Looks like SR20 closes around mid-November on a regular basis.
As an aside, I figured out why my car felt like it was riding on koosh balls. The tires had been inflated up to only 30 PSI at the dealer, so it was super squishy. I reinflated them to between 33 and 35 at work, and it feels so much more solid now.