I have an imac from 2002 whose hard drive couldnt handle all our itunes library, so I purchased a Verbatim external hard drive and transferred all my music files to it in the summer.
Its been working fine until a few days ago when the desktop icon disappeared. I did all the usual restart activities, and plugged its usb directly into the imac (it was on an extension usb multi plug thing).
Try some hardrive software, such as maxblast 5. Buy a 1 tb drive for around $75. The maxblast software works with seagate or maxblast drives, but other brands have similar software available for free download. If the software finds the external drive, do a drive to drive copy before it fails completely. If you don't need to move your hardrive, get an internal one this time with a locking sata cable, and don't forget to install the power switch lead to the internal drive.