I'm having problems with the sound in this game. The sound effects are not synced properly with the on-screen action. There is a noticeable delay in audio between input/ onscreen action/ then finally gunshot is heard roughly half a second after the fact. Also, the music is extremely quiet.

My audio drivers are up to date. I honestly have no idea and would much appreciate any help or info about this. Thanks.

You can go into the Dosbox settings and alter the the [MIXER] settings to the following. Maybe this will help you :

blocksize=2048
prebuffer=75

A reduced blocksize reduces delay in the sound as it's not providing as much means to buffer the sound.

Also I'd recommend bumping up the cycles : Ctrl-F12.