Hi, you do have all the necessary power cables firmly in the 6950... ? (I forgot that once, felt silly)
The adapters that convert DVI to VGA by switching the pins around are not standard. if the "cable I've got at the moment is a VGA cable with a DVI to VGA attachment on the end. " is not using the adapter that came with the 6950 then switch cables and use the 6950's adapter.
re: "A 6950 2GB with a 19" monitor? Really?" OH YES! bring it.
Also, this is a good ref. Note especially beep codes. http://www.tomshardware.com/forum/261145-31-perform-steps-posting-boot-video-problems
p.s. Doesn't everyone have an old PC that can donate a video card or a friend with an old video card you can borrow to swap in instead of the 6950 ?