xxein: In answer to your first question, the Schwarzschild radius is
the radius defined to be the radius at which incoming matter is sucked
into a BH's lair by gravity at c. So unless you are willing to say
that this velocity will increase to c*sqrt(2) as it gets to the
surface of the BH (r=M, M being the radius of the BH as measured in
meters of light), I see no point in discussing a length contraction
under such a scenario.
For your second question, Wouldn't it be the same as if your rocket
flew past a BH at nearly c using the apparent motion of the BH? But
don't get close. Your speed will fall. You will find out that the
gravity of the BH will vectorialy subtract your forward speed just as
Einstein had to conclude and was subsequently found to be true (light
bends double than what was previously predicted). But even Einstein
didn't really know why outside of a mathematic.
Third question: Would it expose a singularity? Well, consider that
light almost grazing 2M (Schwarzschild radius), could be physically
interpreted as light slowing down to sqrt(5)*c at that point. It will
have double the bend that Einstein came up with in his math and
subsequent observation. But I don't know if such an observation with
a fly-by is possible. But I can tell you that the vectors work.