How to get bearable 2D and 3D performance on AMD Radeon HD 6950?

I have had an AMD Radeon HD 6950 (i.e., Cayman series) for a couple years now, and I have tried a lot of combinations of drivers and settings with terrible results. I'm completely at a loss as to how to proceed. The open source driver has much better 2D performance, but it offloads all OpenGL rendering to the CPU.

What I've tried so far:

• All the latest stable Ubuntu releases in the period, plus one Linux Mint release.
• All the latest stable AMD Catalyst Proprietary Display Drivers, and currently 13.1.
• The unofficial wiki installation instructions for every Ubuntu version and the semi-official Ubuntu instructions.
• All the tips and tweaks I could find for Minecraft (Optifine, reducing settings to minimum), VLC (postprocessing at minimum, rendering at native video size), Catalyst Control Center (flipped every lever in there) and X11 (some binary toggles I can no longer remember).

Results:

• Typically 13-15 FPS in Minecraft, 30 max (100+ in Windows with the same driver version).
• Around 10 FPS in Team Fortress 2 using the official Steam client.
• Choppy video playback, in Flash and with VLC.
• CPU use goes through the roof when rendering video (150% for 1080p on YouTube in Chromium, 100% for 1080p H264 in VLC).
• glxgears shows 12.5 FPS when maximized.
• fgl_glxgears shows 10 FPS when maximized.

Hardware details from lshw:

• Motherboard ASUS P6X58D-E
• CPU Intel Core i7 CPU 950 @ 3.07GHz (never overclocked; 64 bit)
• 6 GB RAM
• Video card product "Cayman PRO [Radeon HD 6950]", vendor "Hynix Semiconductor (Hyundai Electronics)"
• 2 x 1920x1200 monitors, both connected with HDMI.

I feel I must be missing something absolutely fundamental here. Is there no accelerated support for anything on 64-bit architectures? Does a dual monitor completely mess up the driver?

$ fglrxinfo
display: :0  screen: 0
OpenGL vendor string: Advanced Micro Devices, Inc.
OpenGL renderer string: AMD Radeon HD 6900 Series
OpenGL version string: 4.2.11995 Compatibility Profile Context
$ glxinfo | grep 'direct rendering'
direct rendering: Yes

I am currently using the open source driver, with the following results:

• Full frame rate and low CPU load when playing 1080p video.
• Black screen (but music in the background) in Team Fortress 2.
• Similar performance in Minecraft as the Catalyst driver. In hindsight obvious, since both end up offloading the rendering to the CPU.

My /var/log/Xorg.0.log after upgrading to AMD Catalyst 13.1. Some possibly important lines:

(WW) Falling back to old probe method for fglrx
(WW) fglrx: No matching Device section for instance (BusID PCI:0@3:0:1) found

The generated xorg.conf. The disabled "monitor" 0-DFP9 is actually an A/V receiver, which sometimes confuses the monitor drivers when turned on/off (but not in Windows). All three "monitor" devices are connected with HDMI.

Edit: Chris Carter's suggestion to use the xorg-edgers PPA (Catalyst 13.1) resulted in some improvement, but still pretty bad performance overall:

• Minecraft stabilizes at 13-17 FPS, but at least the CPU load is "only" at 45-60%.
• Still 150% CPU use for 1080p video rendering on YouTube in Chromium.
• Massive improvement for 1080p H264 in VLC: 40-50% CPU use and no visible jitter
• glxgears performance about doubled to 25-30 FPS when maximized.
• fgl_glxgears still at ~10 FPS when maximized.