the video does look quite stuttery to me, in fact it looks like vsync on but without hitting the vsync rate so it stutters. is this a 30fps or 60fps capture? a good idea here would be to turn on fps display, i think cl_showfps 1 is the console command, also the game needs to be run with hl.exe -console shortcut. iirc the game also has an fps limit of 72 (fps_max 72), i think the game can take up to 100 or so. with this kind of period setup the best way to play is perhaps vsync off and a high refresh monitor.
not sure what stutters while audio files are played means exactly, but if that's about the pauses that happen when a CD audio track starts, thats a normal issue that occurred in PC games.
as for the z-fighting issues, the d3d renderer is unfortunately plagued by that on many a card, so my preferred way to play is with the opengl renderer and putting the GL ICD in the game folder with the multitexturing extensions hexed out so that overbright lightning works again, this costs some performance too of course. issues with missing or flickering decals in the OGL renderer can usually be fixed by playing around with gl_polyoffset.
also FYI there's no glide renderer in this game, the 3dfx support in the menu refers to the miniGL driver.