The wind must be able to pass through where your axle is at the moment. i.e. it must be open to air flow. It should turn, but with reduced efficiency, I think!
Yes: In the original two-half-cylinder savonius design each half-cylinder should have one edge ending at the radial center of the other half-cylinder.
When the blades are opening-upwind and opening-downwind the concave-upwind blade reverses the jet of air that enters one side of its opening (partilly directed by the outer surface of the concave-downwind blade) and reverses its direction - jetting it out the other side of its opening and into one side of the concave-downwind blade, which reverses it again. Momentum is transferred reasonably efficiently, and the air is decellerated, because both blades are retreating from their respective inbound jets.
If you join the two edges of the blade the concave-upwind is trying to spit the reversed jet back into the incoming air, rather than into the windshadow of the other blade. No soap. Instead, the back of the concave-downwind blade directs some of the air that hits it into the concave-upwind blade, which hooks it back and dumps it overboard. Only one reversal, with lower flow to boot. You also get a bit of torque from the lower drag of the convex surface of one blade versus the concave surface of the other. So much less torque than if the air had a clear s-curve flow through the center and along the inner surface of both blades.