FAA doesn’t open or close airports. Local airport management (which may be a municipal government) may open or close a runway or an entire airport based on snow accumulation. Airplanes may choose to use or not an airport / runway which is open based on the reported weather & reported runway condition (snow, water, etc). Different regs apply to private airplanes, including bizjets, than apply to airliners. Different companies and different aircraft have different operating limitations.
All of these lead up to a matrix of different decision-makers making individual decisions under different criteria for different reasons. So the idea there’s some big light switch flipped by one guy using some simple decision criteria is utterly not the way it works.
As to lawsuits, absolutely everybody has sued absolutely everybody else at least once over history. As a rule governments are especially hard to sue. That means the FAA, the weather service, and most airports.
As to how it does work …
As weather deteriorates from fine to simply cloudy the processing rate of the airport is reduced. If the airport is underutilized this may not matter at all and operations continue as normal. OTOH, if the airport is already scheduled to the max, even fairly minor cloudiness can crimp the flow rate enough to matter. Similarly for high winds on a sunny day.
If an airport is expected to become overloaded, or is already overloaded, these folks get involved to trim the flow of aircraft to remain below the airport capacity. National Airspace System
They do this by ordering departure delays for aircraft going there. This starts with close-in airports who’d otherwise be arriving soon and expands outwards as necessary to trim the “arrival rate” several hours in the future to match expected airport capacity then.
This is your classic “sitting around at the gate waiting for ATC approval to leave” scenario.
If delays get long enough, the airline may choose instead to just cancel the flight.
As the weather gets foggier, some airplanes can’t land; they lack the avionics and crew training necessary to operate to the lowest weather. The airport is fully usable to the airplanes and crews that have the right stuff. This land/no land decision is made by the pilots inflight; nobody on the ground decides anything for them. The weather people report the current weather, ATC passes it along, and the arriving pilots consult their personal / company decision tree and proceed or not as they choose.
For flights still on the ground the airline’s planning staff will be looking at forecasts and current reports. There’s no point in going if there’s little to no chance you can land. OTOH, if you’re coming a long way it might make more sense to leave, e.g., Europe for JFK even though its fogged in now and is still expected to be fogged in 10 hours from now when you expect to arrive. You make this decision (and carry extra fuel) confident that you can land someplace in the US Northeast (maybe even JFK!) and get the folks (and the crew and airplane) much closer to the destination.
Historically Southwest didn’t bother with this lower weather capability since it was expensive and they like cheap. So often they were cancelling all their flights to a foggy city while everybody else was going there just fine. Over the years as they’ve expanded into the northeast and gotten newer airplanes they’ve relented and gained some more of the capability everybody else has. Many RJ operators still have this limitation today.
Fog can also limit takeoffs and the takeoff flow rate. Which in turn can back up into the terminal leading to gridlock there.
Snow: Snow has several synergistically bad effects. It reduces flight and taxi visibility. It reduces braking & steering traction on runways and taxiways. It slows all taxi operations. It requires airplanes to be deiced before takeoff. It requires runways and taxiways and ramps to be closed and plowed over and over as long as snowfall continues. It slows all aspects of ground support, from catering trucks to fuel trucks to the next shift of airline / airport workers even getting to work.
But other than closing an individual runway while the snowplows are driving around on it, snow doesn’t really close airports. It just gums them up to near immobility.
Bottom line:
A lot of the cancellations you see nowadays are not because *one *airplane couldn’t go there and operate just fine. It’s because a *hundred *airplanes can’t all go there the same hour and operate just fine. The industry has realized that a gridlocked big airport is a PR and customer service disaster in the making. Far smarter to avoid it by cancelling vast numbers of flights ahead of time and deal with the passengers elsewhere, not all concentrated at a city they don’t want to be in anyhow.
That’s enough for the overview.