Since BG EE is DRM free, does that mean that I can play the game cooperatively with my brother (on another computer in another house) using a single copy of BG?

No, not legally. The Beamdog license is a single user license. Here is the pertinent part:

"This Agreement grants a single user license to the Licensee. The user is allowed to operate only one instance of the software at any time. Abuse of the system will be tracked and in cases of abuse the user account will be terminated."

It has a serial number, so it's not cometely DRM free for the multiplayer side. I don't even know if it's *possible* to play it like that. But I haven't tested it

You definitely can for the standard editions, technically. But GOG's policy is that you can only keep them playing in your house. They shouldn't leave your household.

The serial number is a code that lets you download the classic versions of BG, I believe. Beamdog made those versions available only if you buy the EE versions. They surpassed EA when it comes to slimy moves with that one.

Nope. The only examples I can think of where multiplayer is possible with only one copy is Nintendo DS/3DS games (I think its called Download and Play), and early Blizzard games (the other player needs to install a "spawn" copy).

As far as I know other companies haven't done anything like that. There's an idea for ya, Beamdog. You'll get free advertising as more people are exposed to the Enhanced Editions, and if players with spawned copies want to play solo, they'll have to buy the game from you. You'd probably turn this idea into a dog's breakfast, but I think its worth the attempt.