I used to play battlenet back in the day to get the hellfire torch and anihilus charms that you couldn't get on single player. It was super easy to get high runewords too, so it made getting some of those possible. Anyway, yes, the 3 of you could get just make a game on battlenet and set a password for it. One would create the game and the other 2 could join. You can play solo on battlenet too. The drawback to battlenet is that the maps change every time you start a new game. Also, you can't change player settings (/players3, etc) like you can in single player. Also, yes you could get "banned" for creating too many games in a set time period. It was to cut down on bots or something like that. It happened to me a few times, but the bans didn't last very long, maybe a half hour or so. It was a long time ago so my details might be fuzzy. It doesn't happen that often, so that should not be a deterrent for you. All in all, if you have never done battlenet before you should try it out.
Hopefully they come out with the PlugY mod for D2R and then that will be the end all for me.