Depending on where the cable is severed, the majority of it is likely to just fly off into space. Remember, it's being actively pulled by centripetal force.
However, if it's severed at some point in the middle, say just below the geostationary orbit point around halfway up the cable, then the majority of what falls toward Earth would actually just burn up in the atmosphere.
Only about 100 miles of cable would likely reach the ground, and it would probably weigh about as much as your average power line of similar length.
Indeed. The real question about a Space Elevator is not whether it would be safe in a disaster, the question is whether it would work, at all.
If Space Elevator terrorism is to become a thing someday, it probably won't take the form of threats to cut the elevator. Especially once you have more than one, that's not a very interesting attack (and even if you only have one, the damage would be primarily economic, not physical). The way you'd do terrorism is to sneak something terrifying into orbit, since it's so much cheaper to do that than it is now.
It is something the elevator cars will be designed for, and just by the nature of the elevator we'll have a lot more orbital infrastructure ready to receive the cars. IIRC, much like airplanes at takeoff and landing, there are some windows where it is impractical to recover from a detached elevator that occurs at precisely the wrong moment, but they are relatively small. (Depending on the precise numbers, it may be possible to build a passenger car that has no such windows, but cargo cars will probably just run the risks since it greatly increases their payload.)
I won't promise NOTHING WILL EVER GO WRONG!!!1!, but there are a lot of engineering options to mitigate disaster.
However, if it's severed at some point in the middle, say just below the geostationary orbit point around halfway up the cable, then the majority of what falls toward Earth would actually just burn up in the atmosphere.
Only about 100 miles of cable would likely reach the ground, and it would probably weigh about as much as your average power line of similar length.