you cannot connect a usb hard drive to the esata afaik.(edit:without an adapter)
But as far as usb 3.0 is concerned the usb 3 is backward compatible and a usb 2.0 device can be connected to a usb 3.0 port
The esata to usb adapter wont give any boost in performance.
The usb 2.0 hard drive can be connected to the usb3.0 port normally and the esata using the adapter but none of those will give it a speed boost afaik.
So what would be the best external hard drive solution that allowed me to use USB 3.0 and eSATA?
The Seagate Freeagent Go HD (link provided above) seems to require a Seagate connector cable. Is this true or are there other non-Seagate cables that I could use with it instead?
You can use a USB 3.0 port with your current hard drive but it will be the same speed as USB 2.0 because that's the port the hard drive has. No adapter will be able to change that.
Same with a SATA to USB cable. The connection will always be limited by the slowest part, which in this case is the hard drive. (also I think the SATA to USB is meant for drives that don't have a USB port)