I've had numerous external hard drives go bad and each and every time I've pulled the drive from the case and either slaved the drive into my desktop or I own a Thermaltake Sata HD Dock (http://www.amazon.com/Thermaltake-BlacX-...) that I picked up for around $45, connects Sata to USB, and works like a charm. I work as a Data Center Tech and I've seen external drives go bad more than I can count and every time I've had to pull the drive to save the data. Not sure where you heard pulling it out of the enclosure will render it useless or remove a partition, but yeah that's untrue.
Just recently, I had a 2.5" 320gb Toshiba USB Portable Drive Enclosure I had picked up for $10 bucks from a pawn shop that did something similar to your situation. The drive would spin up for a moment after the initial connection but then would take 15-20 mins to be recognized by the computer, then another 15-20 mins once I clicked on the drive to access the data, it'd do the clicking, spinning, the whole bit. I had a spare drive 320gb 2.5" Sata HD that I pulled from a Toshiba Laptop (w/ a busted screen) which I pulled and amazingly it was the identical drive that was in my portable. I swapped the drive and it worked great for like 5 days then starting exhibiting similar symptoms as before, but yet slightly different. This time the drive would just sit and click over and over, sometimes it would decide to work, most of the time it wouldn't . Needless to say I threw the Toshiba External HD Enclosure in the trash, I slaved one of the Sata drives into my desktop and used the other to repair a Compaq that had a bad drive. Both drives have worked great ever since and luckily was able to keep the most important piece of it all, my data.
Now if it's the drive itself, like bad sectors, etc. you'll still want to pull it out of the enclosure and connect it as a secondary drive into another computer that you can run a HD utility like chkdsk or something similar. If it finds bad sectors, many times it can repair them and tag the sectors to not be used again. If the drive has been dropped one too many times, you may be SOL if the platters are damaged. No amount of salvaging can save the drive. From what you describe I'd take a guess and say it's not the drive itself but instead the enclosure that is causing the drives to malfunction. Hope this helps.