This is one of the reasons why I'm suspicious of camera-only systems, here in Finland. Half the year there's a lot of snow and ice around. Which I imagine means most of the view is "white" and "shiny". Coupled with the dark winters it's gotta be a nightmare to deal with.

do humans drive in it?

Plenty of snow and ice in parts of the US. Hopefully you'll get a chance to try the self-driving for yourself.

Could you use a different spectrum of EM radiation to detect water? There are parts of the microwave band that attenuate the signal by absorption and I wonder if you could use that. The only clue a human driver has in that situation is in the visible spectrum. The lines of the road disappear from view, which can be challenging to see at night.

In theory you could send different frequencies, but then you run afoul of all kinds of potential interference with other systems and other local regulations.

If the LIDAR can sense the road close enough to the front of the car, then it could estimate how far underwater the car is.

There are better ways to measure how far underwater the car is: https://www.landrover.com/ownership/off-road/wading-depths.h..., this assumes you driving over something more or less flat underwater and that car is built for this.

Here the goal is avoiding driving into the water in the first place.

Not necessarily. Depending on angle and water depth, multi-return LIDAR can give you returns from both water surface and the road surface beneath, in the same way multi-return LIDAR can produce returns from vegetation and the ground beneath.