Based on your edited question in response to the comment above, I conclude that given due consideration, you find that there is no special risk of fire from the use of oil or grease on the underside of a car when used to prevent rust.
If you park over a campfire or drive through burning brush, then all bets are off. But let's face it, a little oil or grease adding to the fire would be the least of your worries at that point.
Standard factory or aftermarket undercoating contains asphalt, a very thick form of oil that comes from petroleum, and it is flammable too if exposed to flames. Some companies such as VW use (or have used) lanolin, which is sheep wool grease, for rust prevention inside box beams in the sheet metal that forms their monocoque chassis. It will burn in a fire. All the plastic parts under a car will burn as well if exposed to flame. And let's not forget about the tank full of gasoline or diesel fuel.
To answer your bonus questions:
I live in New England in the northeast of the USA. We get a lot of snow in winter and road crews use salt. This area is part of the "rust belt" where some cars rust quickly from all the salt.
Many people in this area have an annual tradition of taking their car to be oiled before winter snows start. A neighborhood mechanic will put the car on a lift and mix up a solution of motor oil thinned with kerosene. They apply the mixture to the entire underside of the car with a spray gun or with an air chuck that has an atomizing venturi tube. It makes a mess and it makes the car smell like an oil well, but it does help to prevent rust. A few businesses in the area specialize in rust prevention and use a secret formula that is oily but does not smell as bad.
No one applies the stuff to individual bolts; they simply spray everything, everywhere. And unlike undercoating, which would un-balance a driveshaft or a steel wheel, it's OK to use oil there; it's such a thin coating that it won't cause an imbalance.
Yes, some of the oil washes off in the rain and some washes off in the car wash if you get the undercarriage wash. But it doesn't all wash off. A car is usually still a little oily, and therefore somewhat protected from rust, by winter's end. The whole process has to be repeated every year.
It's not a perfect system and it probably contributes to water pollution from oily road runoff, but it does extend the life of vehicles in the rust belt.