How difficult would it be to control the final shape when using subdivision compared to using spline schemes?

Since subdivision is iterative, does that make it inefficient in practice? How does performance compare to a Bezier Curve?

Are each of these steps computed sequentially, or can you just go straight to the last step when rasterizing?

After what point does subdivision stop subdividing midpoints?

How much does the control curve matter? Given any control curve, are we able to obtain all possible shapes within it?

If the intended final shape had both a curved edge and a flat edge (e.g. a semi-circle), would subdivision continue to divide midpoints on the flat edges so long as the curved edge isn't ideal yet/to the choice specifications? or would the flat edge stop to some extent?

Is this done at render time? Or is this usually pre-processing done on the geometry once before being rendered?

Is there some analytical proof out there for why subdividing midpoints using a certain scheme approaches a Bezier curve?

Is there a method behind calculating the best averaging rule in each case, or is it just a guess and check process?