Let's back up a bit. Your Cox bundled-signal (TV, Internet, Telephone) enters your house at the NOKIA G-010G-A...aka Optical Network Terminal...aka ONT. The ONT separates your bundled signal (TV, Internet, Telephone) and sends each signal to a port on the ONT. From the ONT, you can trace via wires to each device.
I assume a Cox tech installed an amplifier in your home because your TV signal out of the ONT is weak and needed a boost. Somewhere between the ONT and the Mini Box should be an amplifier. You said, "There is an AC adaptor connected to the coax" so I assumed this AC adapter is near the Mini Box.
This AC adapter is supposed to provide AC power through a coaxial cable to a distance amplifier. Why this amplifier isn't near the ONT is beyond me...perhaps no available AC outlet...but somebody thought a good location was near your Mini Box. An amplifier could be in your attic or as Tiffany suggested, connected to an unused coaxial outlet.
When you disconnected the AC adapter, the other boxes lost signals. The Mini Box didn't lose its signal because apparently it wasn't getting a signal...hence you swapped it to troubleshoot.
However, this AC adapter itself may be the amplifier and there is no distance amplifier. I wouldn't think this would be true because amplifiers should be bigger and most sophisticated than a blocky "wall wart" (AC adapter).
Is the AC adapter near the Mini Box? Is the AC adapter connected to a coaxial splitter?