How do I generate uniform random samples on the hemisphere facing a point p, and defined with normal vector n?

Answer 1

Uniform and Random are contradictory constraints.
I assume what you wanted was sufficient randomness to ensure that you were not systematically excluding some important point. I assume by uniform, what you wanted was sufficient coverage of the surface that no significant feature would be missed. I also assume that the normal vector is normal to the surface of the hemisphere at the chosen point.
There are any number of methods of generating random numbers. Pick one.
Generate two random numbers.
Use the two random numbers to generate an equivalent pair of angles from -90 to +90 degrees.
The normal through point P should be considered (0,0)
The point determined by the the two angles is also the start of a vector defined by the point and the two directions. This vector is the normal requested.