In game's like Battlefield or Arma, how many times per second is the position of a bullet calculated?
You should use projectiles only for comparatively slow "bullets" like grenades from a grenade launcher or RPGs.
For very fast bullets like bullets fired from a gun, A ray cast (or a ray-world intersection test) is used instead. A ray is fired from the barrel of the gun (origin of the ray) in the direction the character is aiming (direction of the ray). If the ray hits something, the hit is registered to the code handling bullet hits after a delay which is proportionate to the distance of the hit point from the ray origin.
If you take the average muzzle velocity to be say 1 km/s, the time delay can be calculated using the simple
time = distance / speed formula. In most cases, the distance will not be more than 100 meters (unless using sniper rifles). If the delay is more than say 0.1 seconds, you could check in the code handling bullet hits whether the object hit (or its bounding box) still contains the hit point. If it does, the bullet has hit that object.
You could do extensive research and find the muzzle velocity of each gun you have in the game and use it for each gun separately, or you could use an average figure.