Are runway safety areas for both emergency landings and aborted take-offs? Is a runway safety area at the end of a runway provided specifically for safe emergency landing of aircraft using it and are required to abort takeoff just before or after reaching the critical V1 speed? Does it mean that at big international airports with a number of runways, each and every runway is to be provided with its own runway safety area?
Yes, the runway safety area is to provide a buffer zone for aircraft that leave the runway for whatever reason, including rejecting a take-off late or by overshooting on landing. It's not specifically for an "emergency landing": sometimes the emergency might only arise during the course of the landing, for example tyres bursting or pilot error.

If it were only for emergency landings, then parallel runways probably wouldn't need to all have a RSA, because the distress aircraft could be offered the one runway that has one. But in fact every runway needs an RSA, because any landing might suffer a runway excursion.