Does each instance of Diablo require both a unique Classic AND Expansion key?

Yes to download the full expansion set of Diablo II the full game it requires the classic and expansion key, if you only have Classic then you can play only classic. .. though if you only have expansion you cannot play diablo II at all. since they are not downloaded separetly.

Just to clarify, this question was directed towards running multiple instances of D2. Not just the initial instance. Does it check the classic key as well?

IE.

Buy 1 classic key.
Buy 2 LoD keys.

Use the same classic key for both installs and /profit.

Running multiple diablo II in the same computer is now considered illegal.

according to Blizzard, running multiple instances of Diablo II or any bnet game is a bannable offense. It cannot be discussed on d2jsp.