What is going on? Did the heat in the external enclosure damage the drive?
It is impossible to say for sure, as we can't for sure say if the heat is a symptom or a cause. I would tend however to support your thesis, though, as heat affects magnetic properties of materials.
Although it is unlikely that your HD reached any close to the curie temperature, heat can still weaken the magnetic proprieties of materials, as cold improves them. It might be (but this is just an hypothesis) that the magnetic proprieties of the disk surface or of the writing heads have weakened.
Also, heat can have deformed the physical shape of some component, making them less effective (for example a writing head which is now further away from the disk surface).
A test you could do is to wrap your HD in a watertight plastic bag and immerse it in a bowl of crumbled ice (even better: you can mix the crumbled ice with salt, that brings the temperature down to circa -21°C) and repeat the tests there. You might notice an increase of performance.
Incidentally his is a technique that - through contraction of the materials - is also useful to unstuck movable parts (which does not seem your problem, as normally a stuck movable part means no read and no write capabilities at all).
Another common cause of disk failure is vibration. Vibration brings lack of precision in the moving parts, tear of joints, wrong alignments between heads and disk surface, and so on and so forth. In case something is now impeding the disk to revolve smoothly, you would for sure have extra heat generated by both the friction and the increased power used by the engine to keep up the rotation at the same speed. In this scenario heat would be a symptom rather than the cause of your problems.
Why can I still use the drive without any problems other than a slow write speed?
With a metaphor: for the same reasons for which you will go faster on a well lubricated bicycle than on a rusty one. Modern hard drives are smart enough to compensate for hardware problems, so - unless a core component is broken - they will find a way to keep running (this is so because HD's obsolete very quickly, and if they would stop working at the first writing error or corrupted sector, you would be changing your HD every few weeks).
What is going on with this SMART Status and what does that Quick Test result mean?
Unless you find some official documentation, this is a question one might only infer the answer to. You can pick your favorite one: from marketing reasons (so you do not immediately notice defects!) to human mistakes (it's just a bug, it should report "not passed") transiting by design ones ("pass" means the HD is still usable, the test that fails signal the fact a non-essential subsistem is broken)
Should I expect this drive to die on me any second?
Again: you can never know for sure. I have still a 5 Gb unit from the 90's up and running, for example. But consider this: you normally would keep backups of a totally healthy HD because it might - all of a sudden - fail. Now, you have an HD with visible signs of bad health status, heating up like crazy, having degraded performance and failing tests... if I were you, I would definitively hope for the best but prepare for the worst!
Hope this helps, and if you try the cryo trick (the ice thing) I would be very interested to know the outcome of it. Best luck!
Best Answer
The #1 sales reason is video, either your own or copying DVD's (may be illegal in your jurisdiction).
Digital photographs can often fill tens of Gigabytes, but you have to be a real shutterbug to fill a Terabyte.
Another use is off-line partitions for backups or mirrors of your boot partition.
Yet another is keeping on-line ISO's of CDRom's, either for production use (developing a Linux Distro for example) or as an online substitute for actually having a physical disk in the drive to allow children's games to run.