When you run a classic XBox game on a 360, does the 360 automatically take advantage of its GPU to run the game at a higher resolution than the original, if you're on an HDTV? In other words, if I play Morrowind on my 360, will it look any sharper (and specifically, will objects be any easier to see) than if I did so on my classic XBox?
It will upscale the game's resolution but upscaling can only do so much. At the end of the day, the original output only has so much detail, as is obvious from many of the Blu-ray re-releases of old stuff. (e.g. Monty Python looked poor on my DVD edition, compared to other DVDs, so the Blu-ray one can't be great).
Another point to consider is that many Xbox games are not playable on 360. This is the list of all Xbox games that work on 360. As you can see, the last time MS improved the compatibility was November 2007, so this is as far as it will ever go, more than likely.
EDIT: Since a site redesign, the Xbox.com page is no longer available. Here is a wikipedia page for compatibility instead. A recent update to the 360 has actually reduced compatibility.
Playing an original Xbox game on the 360 will upscale the resolution. But that is the only change.
Some people like the difference, other people don't. Most people describe the difference as being "sharper" or "harsher".
This can improve contrast, which might make things easier to see, but again, it can depend on the individual.