How long does it take for a black hole to fully form? 

If time appears to "stand still" at the centre of a black hole (from an outsiders point of view), does this mean that it takes forever for a singularity to form (as far as the rest of the universe is concerned)?

Or does the horizon make this problem go away?

I have heard that black holes have a finite mass but no size and they most likely contain singularities which have infinite mass. How can a finite mass object contain infinite mass? Or am I thinking the question wrongly?