- Pneumatic pusher is the term normally used. The Falcon 9 manual itself refers to them as such. It's not just one pusher. Falcon 9 V1.1 has four: three on the rim of the interstage and one on a quadrapod mount inside the interstage.
- I can't tell how common pneumatic pushers are in stage separation, but pyrotechnics seem by far the most common, and that suggests pneumatic pushers must be rare. Falcon 1 also used pneumatic pushers, and Falcon Heavy probably uses them too.
New Glenn, Blue Origin's rocket, uses pneumatic pushers for stage separation as well. From the New Glenn payload users' guide:
Not to go on too much of a tangent, but stage separation isn't the only place on the Falcons with pneumatic pushers.
The landing legs use them to kick the legs out the first few degrees so that the much much larger telescopic cylinders can extend them all the way down. (When stowed the cylinders are positioned so that they cannot extend even if pressure is applied---they must must be rotated a few degrees out first).
Pneumatic pushers are also used to separate the Dragon capsule from the trunk---e.g., during mission abort---and to separate the two halves of the bullet-shaped payload fairing (which the Falcon 9 manual says uses four pneumatic pushers).