The instruction manuals provide a tip that recommends using the keyboard and mouse simultaneously, saying "the mouse provides fine control for aiming your weapon (allowing you to smoothly rotate right and left) [...]". The keyboard and mouse combination also provides rapid rotation and easier strafing, resulting in quicker maneuvers than when using the keyboard alone.
Depends. If I'm on, say, a laptop or just want to kick it with a keyboard, I would make WASD my forward/backward and strafing movements and arrowkeys turn, and then the up and down arrow keys can be mapped to fire/open.
Keyboard. I wish there were still FPS games that don't really use a mouse. I miss the simplicity. Doom is still the apex of the genre to me. There's something incredibly visceral about it that no other game has managed to capture to the same extent. The first Quake was pretty great too though.
Mouse for x-axis aiming is great, my preferred gameplay setup when playing vanilla. Mods are another beast though, some are damn near impossible to complete without full mouse aiming.