A1) False.  It's *local* search...
B1) False.
A2) False.  Going in the direction of the gradient will maximize, not minimize
B2) False.  It may diverge by overshooting minima
A3) True.
B3) False.
A4) False.  If h1(n) < h2(n) forall n, and both are admissible, then h2 is the
better heuristic and will (potentially) expand fewer nodes.
B4) False.  Best First Search is incomplete and not even guaranteed to find the goalstate.