I'd like to play RoTT2013 with someone who has it on Steam, and I'd like to know if the GOG version can play in an MP game alongside the Steam version.

Thanks ahead of time! :)

Right now you can't because ROTT is now using Steamworks on Steam.

Yep, they switched to steamworks so in order to play MP you need wait until GOG similar Galaxy MP infrastructure in next patch. However all old accounts system still works. I own it on Steam too but didn't noticed any special boost in performance so you still need the same community tweaks to increase framerate in the same critical levels. Other tweaks from 1.5 are subtle not too different on stabillity or optimization from what I tested (had two on the same machine installed).