Is 223 Remington the same as 556mm?

Answer 1

yes, except 556 is written as 5.56 caliber (223)

Actually, no, it isn't. Although the dimensions match, the military 5.56 cartridge differs from the civilian .223 in several aspects, such as the neck angle, casing thickness, and chamber pressure.

A general rule of thumb is that .223 Remington can be fired in a 5.56 chamber, but 5.56 ammo should not be fired in a dedicated .223 chamber.