Extreme weather... What would it actually do? It would prevent agriculture. It would feasibly destroy houses and most public buildings. It would kill the vast majority of your elves, mainly via famine and disease. That last bit is important to remember for the social effects.
Effects on the ecosystem at large? Since we are talking about extreme we can probably say that trees will be gone. I mean, if the trees could survive, it would be fairly easy to create buildings from wood that would also survive. So the landscape would probably be something like prairies or savanna. Grasslands with occasional small trees or bushes.
We can also assume common flooding and extreme erosion. Grasses are actually good at dealing with erosion and lack of rain is a major issue with the Earth analogues I gave, so it would probably be very rich and productive ecosystem but with spectacularly deep canyons or gorges for the rivers.
For animals you'd have antelopes, horses (or zebras), and some sort of bison/wildebeest equivalent. These would be hunted by lions, leopards, wolves. You'd also need scavengers such as hyenas. Very rich ecosystem very similar to one we have all seen in various nature documentaries over the years.
The point I am trying to make is that even with agriculture being impossible there is going to be lots of food outside. Lots more than underground, certainly. Your elves will live in the caves and hunt the animals outside. Maybe ones that come to drink at the river valley their cave overlooks. In short, they will live almost exactly like our ancestors lived before the agricultural revolution.
It would be fairly easy to assume that the weather is worse than this and there won't be rich ecosystem outside but, let us be honest, if the weather is so extreme that grasses cannot survive it, everyone will die. And something will be hardy enough to survive to eat those grasses. Or elves certainly will not survive.
Your elves would start with iron working, large scale organization, and knowledge of architecture. They would gather survivors, maintain order, scavenge all the tools and valuables they can find. The drop in population would cause major social changes (no large kingdoms) but medieval technology should be sustainable, if the elves stay organized.
This works much better if there is a nearby cave system that can hold thousands of elves that they can withdraw into and indeed have withdrawn in the past during wars and such. The end result would be independent elven communities with sizes from a small village to a small town depending on the size of the cave system available.
The elves would have the ability to expand their caves for more space. They would also build small gardens protected from wind and flooding. Eventually they might develop mushroom farming but initially they wouldn't have the space to do it to an extent that makes a difference.
Fires would be an issue because the storms would destroy most trees, so after the initial windfall gets burned or swept away, they'd have to start burning compressed manure collected from outside. While some trees would survive due to protected locations, those would be the exact locations where you'd want to build a garden for fruits or vegetables, or even build a house on. Far too rare and valuable to grow fire wood on.
Given the size of the settlements they would probably be chiefdoms, they'd be run by a chief, probably a descendant of the leader who founded the settlement, but probably with the approval of "clan elders". Sometimes decisions or chiefs might be discussed publicly and even voted on.
Religion would probably be shamanism of sorts. people would probably mainly want good luck with hunting and avoid bad luck with weather, famine, disease, so a combined spirit talker, witch doctor, wise man like a shaman would be quite appealing. Specialized priest or organized religion would generally require larger societies. And more wealth and leisure time than shell shocked survivors could provide. Besides after the weather suddenly turned against them people would quite reasonably blame nobility and priest for angering the gods. They might be quite devout and religious but organized religion with priest and dogma would probably not be that appealing.