I use Av most of the time, because it gives me control over the aperture, which determines the depth-of-field. I often use partial metering and meter the brightest part of the area and apply some + exposure compensation. Remember that the camera will set the exposure so that what the meter reads will be average gray in the image. So if you meter on a bright part with the partial meter, you should use for example +1 1/2 to +2 exposure compensation to make sure that the bright part will be 1 1/2 or 2 stops brighter than average gray.
When I use flash as my main light source, I set the camera to M, the shutter speed to 1/200 and I set the aperture to get the desired DOF. When using flash, the shutter speed isn't very important. On the 10D, 1/200 is the fastest shutter speed that can be used when using flash. See Flash Photography with Canon EOS Cameras
for everything you want to know about flash photography.
Show blur: Use Tv, choose shutter speed 1/60 or slower (depends on the amount of blur, speed and distance of the subject). You might need IS or a tripod for really slow shutter speeds.
Freeze action: Use Tv, choose shutter speed at least 1/200 (again, depends on speed and distance of subject).
Control DOF: Av, because aperture setting controls DOF.
Green square?! Never!!!
I don't want the camera to think and make decisions for me.