Here in Northern Kollyfornya, They have 'Mandatory' chain laws when it snows on the mountain passes This means that if you have a 2wd vehicle WITHOUT mud & snow tires, you need tire chains. Mud & snow tires will get you through. This is considered "R1", and is the most common restriction. "R2" means ALL 2wd vehicles, M&S tires or not, need chains, and 4wd with M&S tires are OK. In 30 years of living here, I've only seen an "R3", or only 4wd with chains allowed, restriction happen once, and that was several years ago. As SST said above, Caltrans closes the road first. As for studded snow tires, they are allowed from Oct, 31st to May 1st, and yes, they do tear up the road something fierce...
AS for chaining up the ST, the owner's manual sez "No chains"...