I figured out how to get Diablo 2 running on my system, and it would be very valuable to update existing tech support documents with this information.
Graphics issue related crashes are not solved by run-as-admin. Instead, if you set your desktop to 1366x768 (or other resolutions that were common at the time D2 was released), then the game starts up fine, and displays videos.
I am guessing this has something to do with what happens when Diablo is trying to change the video mode on your computer.
I also believe this is far better of a solution than the current recommendation of -w for windowed mode, as you do not get the cinematics in windowed mode, but you do in fullscreen.
In addition, this allows for perspective mode, which is also disabled when playing in windowed mode.
In addition, this removes the annoyance of accidentally clicking outside of the window.