Yes, but it won't damage the gun in any way.
Yes. Recommend using a snap cap.
If you are asking about a "springer firing system" The answer is NO do not dryfire it. When you shoot a single cock springer rifle or gun with a pellet in the chamber, there is a moment where the piston actually slows down just before the pellet is fired down the barrel. If you have no pellet in the chamber the piston slams against the end of the chamber and this will eventually ruin the chamber. Dry firing a Co2 gun will not hurt it. You will just waist Co2.
To Dryfire any gun means to fire it without any projectile in the chamber. No BB's or pellets. For most airsoft guns dry-firing is harmful and may lead to breaking some internal parts. Gearbox can get cracks after much dry-firing.
Yes. The firing pin is intended to make contact with a cartridge primer. When this doesn't happen, and it "lands" on an empty chamber, this causes resonance in the firing pin which can increase wear and eventually break it. A much better alternative to dry firing would be to use either spent cartridge casings or the "snap cap" dummy cartridges which you can find at most gun shops. ***** Agree with above. In addition, dry firing many rim fire weapons may dent the mouth of the chamber, resulting in misfires.
Most spring Pellet guns should not be dry-fired. When you shoot a spring powered air gun or rifle, the spring compresses the air in the airtube and the compressed air forces the pellet out of the barrel. The compressed air also slows down the spring piston due to the air being compressed. In this way the piston doesn't slam against the end of the chamber. When you dry-fire the rifle without a pellet in the breech the piston has no resistance in the chamber and slams against the end of the airtube. Eventually it will ruin the air tube and the rifle or gun. If you dry-fire a C02 gun nothing will happen no damage will occur.