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Not having much luck trying to run Quake Engines with the GOG installation of Quake The Offering ... highly distorted Graphics with Quakespasm ... Blue Screen of Death with DirectQ ... Page Fault / Sysaudio error. This particular setup is AMD Athlon 64 with Ati Radeon HD3850

Weird, never seen anything like that.
Try launching QuakeSpasm with "-novbo -noglsl" command line options to disable some changes that are more demanding on the OpenGL driver.
Another engine to try is Mark V, http://quakeone.com/markv/ , there are several variants (OpenGL, DirectX, software).
Did you accidentally launch with the "-art" option?