Well, I've finally decided to fully tackle Daggerfall's main quest and while searching for the evidence The Underking wishes for me to search for in castle Faalem, I've run out of leads. I've explored the place from top to bottom and still cannot find what it is I'm looking for, so I'm stuck. I finally caved and decided to try and utilize a teleportation (to the quest target) cheat, yet even though I've edited Daggerfall's Z.CFG file to include a "CHEATMODE 1" line, the cheat, nor any other cheat associated with the cheat mode, will not function. I'm not sure if it's a computer-specific problem, but my PC is running on Windows 7 and is 64-bit. Can anyone please help me... either with getting through Castle Faalem or activating the cheat mode?

This is the quest about the missing/dead prince, right? If so I'm having the same problem... 'cept I haven't tried a cheat mode as I didn't know it existed.

Yes, it is that quest.

When I took this quest recently, it was a letter (a good one) sitting out in the open in the quest location (in this case, I had to climb to the top of a tower that housed a staircase, it sat on the floor. Due to randomization that may or may not be useful.) It was not guarded by a boss or anything, although i'm not sure if that sort of thing is randomized too. It appeared as a curved, light brown paper/scroll icon as you might imagine it would look, about the size of a normal loot pile, maybe slightly smaller.
Also, I play on Win 7 too and the quest item cheat has always worked for me. I'm not sure what to suggest. I didn't have to edit the config because I downloaded the preset dosbox bundle linked to on the uesp wiki - it's seems to be fully patched and set up.
Good luck and happy holidays.

Regarding the inability to use cheats, are you running an up-to-date copy of the game? As far as I know, cheatmode was introduced with the 2.12 patch (or something like that, can't remember the exact number, it was the latest one anyway).

You didn't say what command line you used to start the game. should be

fall z.cfg

This quest always leads to Castle Faalem, but the letter is at a random quest object location. In cheatmode, the ][ keys can cycle through the locations. Rarely, but possibly, you may see the quest object not spawn at all. To fix this you have to reenter the dungeon (but at least the quest isn't broken)

FWIW, one time through this quest, the letter was on the top of the column in the huge hall, which can only be reached by levitation. There's a brazier there, so it's nicely illuminated, and I could see it from one of the balconies around the hall., and just levitate and float over. If your character is relying on climbing instead, you wouldn't have got there.

I don't remember if this quest has a time limit, but I've had the "quest object" disappear on other quests because the time expired and the quest ended in failure.

... was this always the case, or did one of the patches "fix" it?

I remember this quest. I found it once in that section of the dungeon where you have to do the awkward crouch/climb/hop through the interior window to find it. I recall finding it inside one of those purple coffins that is shoved open with a decaying body laying inside, so you might have missed seeing the paper on top of the body. Also, I had to climb the lid of the coffin like a ramp to see it.

As to I. Both time, I remember entering the castle and the letter was right in front of me when I enter, guarded by a ghost.

http://www.uesp.net/wiki/Lore:Night_Falls_on_Sentinel