No, but it determines how well you can steer. Especially in sharp turns you usually have to shift down/up. If there's an obstacle (perhaps a person), both hands on the wheel will give you more room to steer in both directions. You'll be much more limited with one hand.
I (and many others) do. After a traffic light on a large intersection you will have to (or at least should) shift to second gear with most cars before you're clear of the intersection.
It really depends on how you one-hand: arms locked at 12 o'clock, leaning toward the center is a terrible position, but left hand at 9/right hand at 3 (for RHD cars) is perfectly fine for steering response.
