Will free-fall object into black hole exceed speed of light $c$ before hitting black hole surface?

To answer your question you need to be clear what coordinates you're using. If you use coordinates that are co-moving with the rock falling into the black hole then the rock will see the event horizon pass at the speed of light.

External observers, using Schwarzchild coordinates, will see the rock slow down as it approaches the horizon, and if you wait an infinite time you'll see it stop.

External observers obviously can't comment on the speed of the rock after it has passed the event horizon because it takes longer than an infinite time to get there. If you use the rock co-moving coordinates then you can ask what speed you hit the singularity and ... actually I'm not sure what the answer is. I'll have to go away and think about it.

Incidentally http://jila.colorado.edu/~ajsh/insidebh/schw.html is a fun site describing what happens when you fall into a black hole.

Nope, it will just fall in a reasonable amount of time (if you go with it, but watch out for tidal forces!), or take forever to fall in (if you are watching from outside).

Also, if I may be so bold as to suggest doing some quantum mechanics instead of kinematics while you are there, you could probably lock down some funding no problem.