Yes, the inhabitants remember very well what happened, and have it well-documented in their history books. In the aftermath of the war after everyone migrated underground, the races made treaties governing the use of magic and other things to prevent another apocalypse, and also since their new home is more complex/fragile than the surface, they need to keep people in line to prevent some spellcaster from accidentally collapsing the whole place on top of them.
Still working on the details of the war and exactly how the surface was destroyed, but for a while all the cultures pointed fingers at each other, but after so many years those feelings have mostly died down. The real instigator of the war was a mysterious powerful race dwelling in the deepest caves that no one else is aware of, or if anyone is aware of it they're in a minority and aren't believed. (Kinda like "the Others" in A Song of Ice and Fire, but even more secretive; these "others" are so unknown they don't even have a small place in the mythology or legends of the culturess.)
So it will feel a little similar to Song of Ice and Fire, or the Midnight campaign setting, in that there is a powerful mysterious overarching "great enemy" to the setting. Except these enemies don't make themselves known because at least their plan has already succeeded. Still fleshing out these "enemies" and why they forced everyone to move underground. Perhaps these enemies are a cursed race who were punished by the gods to never see sunlight again for some archaic misdeed, and now they want everyone else to suffer similarly, or something along those lines. Not sure yet.
Yeah Morph3us, I will definitely look into Dark Sun for ideas of the surface. Except in this setting it's not even physically possible for people to live up there, since the air and water are toxic (think magical version of radiation perhaps). Of course, undead and demons and such like it just fine. I haven't given much thought to an actual layout of the surface just yet, as the main focus of the setting is the Underworld.
One big hurdle I have though is maps. It's easy enough to map out the caves from a top-down perspective, but caves are in 3 dimensions. Not sure how I could represent that in a 2d way. Modern speleologists have sophisicated 3d mapping software for mapping cave systems, but I'm not sure how people in a fantasy setting would accomplish that. This setting is separated into 3 layers: Shallow Lands (humans, halflings, orcs, various half-breeds, a couple new races), Krogaunud (dwarven for 'twisted tunnels') aka The Honeycomb, a no-man's land of bewildering twisting tunnels with few resources, and the Deep Lands (elves, dwarves, gnomes, various half-breeds, a couple new races). Hard figuring out how to represent 3 dimensions. I gave up on any notion of making a map of The Honeycomb since its very nature is too bewildering to map...