I have a similar problem with the Seagate Expansion Drive 3Tb, but the problem is not that it goes "offline" but the drive persists in the "My Computer" list when the laptop is put to sleep and the Seagate drive disconnected. Moreover, prior to hibernating the laptop, you cannot eject the Seagate drive from the USB tray widget, "Safely Remove Hardware and Eject Media".
As a result I nearly had a heart attack when this first happened since I had plugged a *different* hard drive in prior to waking up the laptop, and to my surprise, a 1.5 Tb generic drive was then being identified as the 3Tb Seagate Expansion Drive! Before I realized this, I had attempted to transfer some files to the misidentified drive, and the copy dialog appeared to be transfering the files, but eventually started spewing error messages. That flurry of error messages eventually got me looking at "My Computer", and that is when I realized the drive list was completely wrong as a result of the weird behavior of the Seagate Expansion Drive.
This persistance-on-sleep behavior exists even when you check the drive property (from the device manager) to make it "Quick removal (default)". This drive policy, "Disables write caching on the device and in Windows, but you can disconnect the device safely without using the Safely Remove Hardware notification icon." But not with the Seagate Expansion Drive. It doesn't work.
Like you, I found the only way to safely disconnect the Seagate Expansion Drive was to shut down the computer fully and then disconnect the drive. This makes using the Expansion Drive with a laptop a real pain-in-the-neck since most of the time I only just put the laptop to sleep to save time.
