Is that not also an air rifle caliber?
Close. You are thinking of the standard .177 cal, but they are available in other calibres as well.
Yes. .177 is the most common caliber for air rifles. Probably because that is also the diameter of most BBs. I guess .22 is next most popular. 5mm probably should have been the most common but it never really caught on. There are of course various heavy caliber specialty air guns.
.17 caliber never became popular in America. I guess the .17 Hornady Magnum is the most common. Yes it would be deadly up close. The .17 Remington centerfire never really caught on either and I think is now discontinued.
Caliber just refers to the diameter of the projectile. You can get air rifles in .50 caliber and above, and below.