Depends on where you re starting from. At the headwater it will be too shallow even in winter to kayak. If you re starting the lower southern section you certainly can though the increased flow makes it more hazardous.
Yes I m pretty sure you can. It gets colder in the Winter, and of course the days are shorter, but it doesn t freeze or anything. It s colder if you fall in the water, but not so cold as to be a life-threatening, it s just unpleasant.I have gone through Gurneville (a popular resort town on the Russian River) in the Winter and I see people sunbathing on the beach and playing in the water! So I m pretty sure you can bring (or rent) a kayak and go down the river.