I was able to figure this particular error out. It was happening because I had not yet called wglMakeCurrent() on the created context. The documentation was unclear, because it is written primarily for those using D3D so I didn't realize I had to do that. I thought you just had to create the context before calling GPA_OpenContext() but apparently it must also be made current.
I still have not been able to get any information back from the counters, though. I have only enabled one counter, GPUTime. Every time I query it, the value is always 0 no matter what I have tried. I know there should be time in the API call I am trying to profile. The status from all of the calls always returns OK so I'm not sure what is going on there. I am running on Windows 7 and the device ID of my 7970 is 6798.