Since I've registered both my CD Keys with battle.net, I can digitally redownload both Diablo II and the expansion. Do I need to download both to play the expansion, or is downloading only the expansion by itself sufficient?
You need to download both to play the expansion.
Yes, both. I can vouch for this as I had to do it not but 2 weeks ago. Tried the expansion first (thinking they might have been nice and merged them and was unable to even install with downloading the regular first.
Both are fairly small downloads, I think I was playing within 2 hours on a mid level DSL package in the US.
The expansion is as it names indicates an add-on for the original game, not a new game based on the same engine. So yes, you need to download both.
The expansion is meant to be installed upon an already-installed copy of the original game, so you will need to download both, set up the "base" game, then install the expansion on top of that.