Yes, as I though, xorg segfaults there, and it seems to be a bug.
Relevant section in your log file:
14.634] (EE) fglrx(0): Failed to open CMMQS connection.
[ 14.634] (EE)
[ 14.634] (EE) Backtrace:
[ 14.657] (EE) 0: /usr/bin/X (xorg_backtrace+0x48) [0x7f4b7d1bfc78]
[ 14.657] (EE) 1: /usr/bin/X (0x7f4b7d017000+0x1ac969) [0x7f4b7d1c3969]
[ 14.657] (EE) 2: /lib/x86_64-linux-gnu/libpthread.so.0 (0x7f4b7c114000+0x10340) [0x7f4b7c124340]
[ 14.657] (EE) 3: /usr/lib/x86_64-linux-gnu/xorg/extra-modules/modules/drivers/fglrx_drv.so (swlDrmFreeSurfaces+0x42) [0x7f4b781dac72]
[ 14.657] (EE) 4: /usr/lib/x86_64-linux-gnu/xorg/extra-modules/modules/drivers/fglrx_drv.so (xdl_xs115_atiddxDriCloseScreen+0x14d) [0x7f4b781a1b7d]
[ 14.657] (EE) 5: /usr/lib/x86_64-linux-gnu/xorg/extra-modules/modules/drivers/fglrx_drv.so (xdl_xs115_atiddxDriScreenInit+0x8eb) [0x7f4b781a121b]
[ 14.657] (EE) 6: /usr/lib/x86_64-linux-gnu/xorg/extra-modules/modules/drivers/fglrx_drv.so (xdl_xs115_atiddxScreenInit+0xf80) [0x7f4b7819ac50]
[ 14.657] (EE) 7: /usr/bin/X (AddScreen+0x71) [0x7f4b7d06cd41]
[ 14.657] (EE) 8: /usr/bin/X (InitOutput+0x3c8) [0x7f4b7d0adce8]
[ 14.657] (EE) 9: /usr/bin/X (0x7f4b7d017000+0x5975b) [0x7f4b7d07075b]
[ 14.658] (EE) 10: /lib/x86_64-linux-gnu/libc.so.6 (__libc_start_main+0xf5) [0x7f4b7ab53ec5]
[ 14.658] (EE) 11: /usr/bin/X (0x7f4b7d017000+0x44e7e) [0x7f4b7d05be7e]
[ 14.658] (EE)
[ 14.658] (EE) Segmentation fault at address 0x8a0
[ 14.658] (EE)
Fatal server error:
[ 14.658] (EE) Caught signal 11 (Segmentation fault). Server aborting
According to that bug page in theory some beta driver between 13.12 and 14.4 should work, though probably it's hard to find them as they aren't listed on amd.com anymore.
I checked out Phoronix what beta drivers they talk about and I only found some 14.1 beta. This is the one you should try as it supports the kernel which Ubuntu 14.4 has. Although I have no idea how to find it. Google it, maybe you'll find a mirror somewhere. The file name is probably amd-catalyst-14.1-betav1.3-linux-x86.x86_64.zip, though there were other variants maybe.
CPU: AMD Phenom II X4 810 @ 3120MHz | RAM: Kingmax 2x2GB DDR2 800 @ 833MHz| MoBo: MSI K9A2 CF v1.0 (BIOS: 1.D)| GPU: Asus HD 6850 1024MB (DirectCu) @ 835/1135MHz | Display: L24FHD | PSU: PC Power & Cooling Silencer 750 Quad | OS: MS Windows 3.11 Pro x64