Am surprised no one yet cited Elon saying "this is fool's errand" because of price, complexity, etc. I think LIDAR will improve significantly the safety of vehicles (autonomous or not) because it can definitely see better and deeper, even in damning weather conditions (rain, fog, haze). Alas, the price of these devices has to come down at least one order of magnitude. Maybe even two. Still, I am really thankful that…
As far as I’m aware LIDAR cannot see though fog, as the light is dispersed, and even light rain might reduce range significantly. Tesla cars have radar which can see through any weather condition and detect transparent surfaces, invisible to LIDAR.
Simply depends on what wavelength of light you use.
Water-absorbing frequencies are nice because the atmosphere then shields most light, giving you nice SNR from your laser illumination. But better sensors could work around this, using other frequencies that can 'see' through fog.
It's certainly a technological limitation of current systems, but it's not an inherent limitation.