I start up nethack, then I enter my name and roles, and a window opens up saying "oops.", then another saying "Program initialization failed", then "Report the following error to"email@example.com", then "Cannot open dungeon desc. from file "C:\Users[MyName]\Desktop\nhdat" file!" and closes, leaving a document called "paniclog" which repeats "3.4.3 20140306: panic Cannot open dungeon description - "dungeon" from "C:\Users[MyName]\Desktop\nhdat" file!" three times. Is there an alternative download that I can get? I have already re-downloaded it three times already.
I was able to replicate the same problem you have by only having the Nethack executable with the keymapper's .dll's on a folder.
As stated on the download page for the Windows port:
Promise to read Step 2 before telling us the game will not start. Thank you.
Step 2: unpack.
These files can be unpacked with any standard unzipping utility, which should be used when you have set your current directory to the location where you want to place NetHack. Some people have attempted to run the game from inside an unzip utility - this will not work. You must extract the contents of the .zip file to a new folder and run the game from that folder. Do not remove (or copy) the executable from the folder (you can create a shortcut to it instead).
The problem is most likely that you didn't read the manual. Just unpack the archive to a real folder and run Nethack (alongside a plethora of other files) from that folder.
Have you tried reporting it to the email address suggested? It's not likely to yield an immediate fix, but you never know, and worst case, they might fix the bug (if it isn't already fixed) and release a new version one day. (Don't hold your breath though, it's been years since the last release)