I know absolutely nothing about air rifles.
Do you need pellets for all air rifles, or can you use regular bullets? As in a .357cal, can you use .357cal projectiles like one does in a regular pistol?

No.

~4,000 psi v. ~ 40,000 psi.

But, it's a good opportunity ta test your squib removal techniques.

Air rifle "caliber" denotes the size the barrel is drilled for, but as Fubarski said, the power level is less than a firearm of the same barrel size - so the projectile is necessarily lighter, velocity much less.

Mike Holmes