Can a separated ground rod be driven away from other ground rods?

Answer 1

Yes, in fact the code requires two 8' copper ground rods. One under the meter and one 6' to the side of the meter.

The requirement for 2 ground rods is a local requirement and is not required by the NEC. I believe it to be appropriate and is required in my jurisdiction.

Your question seems to involve a separate grounding system, not the one for your primary service. In this case all ground rods are required to be at least 6 feet apart and any 2 or more grounding systems that serve the same structure are required to be connected. From a practical standpoint this means that anything that needs to be grounded should be grounded to your primary grounding system. I wouldn't bother with a separate system.

This requirement is often violated by phone and cable utilities, which will often drive their own ground rod, often only 2 or 3 feet long and smaller in diameter than the NEC requires, to ground their systems. I wouldn't argue with this.