Yes, that's the coin battery. It retains BIOS settings when the computer is shut down. This may be why it keeps changing from ATA to AHCI when you shut down. The battery will not effect your external drive or drive detection.
I'd replace the battery, go into the bios and "load optimized defaults", then change what ever settings you need to, save them and reboot.
If the drive isn't spinning, that is a different problem. It sounds like it's not getting power. Be sure both data and power cables are attached to the drive. You can always try to run the drive inside your PC just to eliminate the external enclosure as a problem.