No, the battle.net version was made for windows. There is a dos version, but thus far it´s not included as an extra.
The DOS version also uses the controls for WC1, which is pretty dated and clunky, even back when WC2 BNE came out.
No, WC2 always had the same mouse controls from original DOS release to later BNE. The BNE release was primarily to bring the game to be played on the web without an IPX wrapper. There was some small polish to the original game, maps, sound, and the proper shared vision was a good thing. But pretty much was the same game. But there were still bugs, in fact I think there was a new one even, which kind of makes the DOS release still desirable. And I doubt the gog release did anything to address bugs.