1. Yes, provided it's a supported controller. I have a wireless Xbox 360 controller and it works perfectly.
2. Game seemed to work offline during the beta. Have not tested the release. As for the game mode, one of the DLC's that will be coming out will be a helo crash simulation. After the crash, your squad will be hunted by some sort of more advanced special force. I believe all game modes are both solo+AI and coop capable.
Hope that helps