Terahertz (THz) emission spectroscopy and microscopy are applied to investigate the electron and lattice dynamics of Ga0.8In0.2N/GaN multiple quantum wells (MQWs). The THz emission consists of three distinct, differently timed signals, whose physical mechanisms are attributed to i) laser-induced ultrafast dynamical screening of built-in bias electric field in MQWs followed by ii) capacitive charge oscillation of the excited carriers and iii) the coherent acoustic phonon (CAP)-driven polarization surge at the discontinuity between the GaN capping layer and air. These multifunctional optical responses show strong dependence on the quantum well width and photon energies. The temporal separation between the first and third THz pulses corresponds to the propagation of the CAP across the GaN capping layer of the MQW structure, whose thickness can thus be determined with 10 nm precision.