In the quasi-two-dimensional (Q2D) electron gas of an InAs channel between an AlSb substrate and superconducting niobium layers, the proximity effect induces a pair potential so that a Q2D mesoscopic superconducting/normal/superconducting (SNS) junction forms in the channel. The pair potential is calculated with quasiclassical Green's functions in the clean limit. For such a junction, alternating Josephson currents and current-voltage characteristics (CVCs) are computed, using the nonequilibrium quasiparticle wavefunctions which solve the time-dependent Bogoliubov-de Gennes equations. The CVCs exhibit features found experimentally by the Kroemer group: a steep rise of the current at small voltages ('foot') changes at a 'corner current' to a much slower increase of current with higher voltages, and the zero-bias differential resistance increases with temperature. Phase-coherent multiple Andreev reflections and the associated Cooper pair transfers are the physical mechanisms responsible for the oscillating Josephson currents and the CVCs. Additional experimental findings not reproduced by the theory require model improvements, especially a consideration of the external current leads which should give rise to hybrid quasiparticle/collective-mode excitations.