Same problems here.
Any USB memory device (whether flash drive, card reader, ext HD) will eventually get spit out by my Mac if plugged in long enough. (And the definition of "long enough" seems to vary greatly depending on the device. Some things get spit out quickly, other things take hours.) Then my Mac will chastize me, telling me I've improperly ejected it. So far, I've been unable to fully Time Machine back-up my system, because my Mac keeps spitting out my back-up! Super lame...
Firewire drives seem to work for me... sort of. I have trouble getting my Firewrie 800 drive to simply be recognized by the system. I have to unplug it, and plug it back in several times before it eventually will be reconized by the system in the first place. The Firewire 400 seems the most reliable, going for hours. Eventually, access to data on the 400 seems to lock up, and I need to re-boot my whole system. But there might be some other problem going on. Not sure. But all of this started since Moutain Lion. I sure am not a fan of Moutain Lion...
But, yeah, right now, I really need to back up my system without my ext HD being rejected by the OS. That's a huge issue.