Now you are starting to understand why lighting is so important even when shooting on video. Since you are indoors, you have complete control over the light, so there's no excuse for not adding some, even if it's some cheap construction lights.
I picked up some paper lanterns at Ikea (6 bucks each). Not only do they make a nice fill light, but if I use lower wattage bulbs they could also do well to boost the overall ambient light of a room without casting hard shadows. The dynamic range of video is much smaller than film so you have to balance your light and dark areas to compensate, i.e. add some light to the shadowy areas or else all that detail is lost.
No matter how you look at it, to get what you're after you are probably going to need to add some light sources or else the resulting image won't be favorable. If you don't want light in the room, what about taking some 1000W construction lights and put them outside, aiming through a window? Add some sheers, and there's your diffused light. Add some blinds, and there's some texture to the light. But note the key word here: "Light". The camera can only see light reflecting from objects. If there isn't much light reflecting, the camera isn't going to see it very well without gain boost and slow shutter (more exposure time). And as you've noted from still photography, slow shutter and wide aperture requires a steady camera or else the image blurs. Same thing happens with video cameras (and film).
EDIT: as for focus, the entire reason I switched from an old Proscan Hi-8 to a Canon XL1s is the Proscan didn't have any manual focus ring and the autofocus was all over the place, especially in low light. It would focus on whatever was in the center of the frame, so if my subject was out of center or if I had any dramatic perspective, it would shift and "hunt" for focus and mess up the shot. With action scenes, by the time an autofocus finds the subject, you've probably missed the action. Blocking the actors in the scene and using manual focus is a better solution. Of course, my experience is still somewhat limited since I haven't used high definition cameras yet, but I suspect the principals are similar.