Hi Roy,
I took the same road on Xmas morning 2 weeks ago. The closed sign for the tunnel it's for heavy vehicles, not cars. At least in my case when I was traveling through VS the signs were showing that the tunnel was closed and than the picture of a truck followed, so it was open for cars (light motor vehicles).
The tunnel costs 45 (or 50) CHF both ways, or around 25-30 if you do it one way, can't remember exactly how much I payed.
But, to reach the tunnel you will need to drive around an hour uphill in the mountains. When I was driving it was snowing and there was snow on the road and also ice. Temperature on this track to the tunnel was -8C.
If you have winter tires and FWD, you'll be OK. As you've never driven on snow and ice before, the rules are
-never press the brake suddenly nor to keep it pressed for longer periods, be very light on both the gas and the brake
-no sudden movements when turning, never press the gas pedal when turning
-don't go more than 30-40 km/h
-if you start to slide, let go of the pedals, no braking, no gas, leave the wheel unless you are on a collision track or going the left lane, in that case just break the slide with turning opposite of the slide
-if you get stuck, try with turning the wheel and going forwards and backwards
Follow this and you'll be fine. There were few cars (I think RWD) that were stopped on the road and needed chains. I experienced only one short slide on ice with M+S ALL SEASON tires and 4WD (well, all wheel drive, 4x4 on demand) but it was very short and going opposite of the slide fixed it immediately. But my tires are not the best for that kind of conditions.
Here's a photo of the track before the tunnel:
The road was full of ice and as I didn't have ideal tires I was going 30-40 km/h. Most cars (even FWD) were over passing me at 50-60 but probably had good winter tires.
The great thing is when you pass the tunnel in around an hour you will see this from your car: