Are US pilots permitted to carry & use their firearm in countries where guns are illegal? On the question of Are pilots armed?, one thing this made me think is: for the few that are armed, are they allowed to bring their weapon into, for example, the UK?

What is the procedure? Does the gun stay in the cockpit or does it need to be checked by security once on the ground?

Presuming a pilot had to fire a gun over EU airspace, is he breaking the law?
Once you land you have to abide by the laws where you are. In the US federal laws (if there are any), when flying in US airspace.

Most US laws about airports and guns, only apply to the secured space. (Inside security), not the GA ramp or anyplace else.