Do I need to download Diablo II and the expansion seperately from battle.net to play?

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.