Which term identifies the half-reaction that occurs at the anode of an operating electrochemical cell?

Oxidation reaction.
Cathode on the other hand is reduction.

Spontaneity tell us the direction of the reaction, but not how fast it goes. gibbs free energy brings together enthalpy, entropy and temperature in one easy formula. g = h - ts. and, finally the sign of delta g is always negative for a spontaneous reaction.