The last liquid-fueled ICBMs in the US Air Force inventory were decommissioned and repurposed as space vehicle launchers in 1987.
ICBM's are equipped with safeguards to prevent accidental detonation. The warhead cannot be armed until an accelerometer detects a certain velocity. The safest course of action would have been to leave the warhead alone when the missile exploded.
When Happy is removing the bolts from the warhead, the torque wrench is set to the opposite motion, which would tighten the bolts.
When Cabe, Happy, and Doc are driving to get the neon gas, one of the camera shots shows the automatic transmission shifter is in Park and not in Drive.
The plot centers on dropping a hand tool, which ruptures the missile's fuel tank. The dropping of tools is a well understood problem; that is why in practice tools are tethered to prevent the drop. A gear head like Happy would know this.
The USAF Strategic Air Command was disbanded and absorbed other various Major Commands in 1992. When a new Major Command with the same basic structure and function was reconstituted in 2009, it was renamed Air Force Global Strike Command.