I want to tackle the unknowns first, and the biggest for me is the spreading of frost
Below is a 100x100 grid of floats with 2 points marked as the source of frost. What you see are spherical gizmos to help visualize what is happening, where the radius of each of the 10,000 spheres is representing the 0 to 1 value in the array at that point in space
I figure I would run this "spread the frost" function followed by the "apply the heat" where fire would basically subtract from it. This is running at 1 update per frame and takes 0.0022s to process, to slow things down I would probably only have it update 10 times a second. As it's a nice 2D array of floats, I can easily use it to determine damage to buildings / freezing of things by just sampling it with X/Y coordinates.
As for how to visualize this in the final game, I am thinking a mesh where each vertex height is the frost intensity and it pokes through a flat ground plane. If that looks janky then maybe some marching cubes or some shader magic (which i've never done before, but how hard can it be? ;) )
And I may use a perlin noise to make it spread in a more organic/interesting way where the value in the 2D perlin noise dictates how quickly frost can spread on it, maybe that will be visible to the player (frost travels at different rates through water, stone, dirt, grass) or it may be hidden