When I try to start Half-Life in full-screen mode, I get a black screen for a second and then the game crashes.

I then tried to run the game in windowed mode and everything worked flawlessly.
I had to change ScreenWindowed from 0 to 1 in ~/.local/share/Steam/SteamApps/common/Half-Life/hl.conf to get the game starting in windowed mode.

My graphics card (Intel Mobile GM965) is pretty bad and old but it's more than enough to run the game comfortably. Running the last drivers.

Hardware info:

Processor Information:
    Vendor:  GenuineIntel
    Speed: 1467 Mhz
    2 logical processors
    2 physical processors
    HyperThreading:  Unsupported
    FCMOV:  Supported
    SSE2:  Supported
    SSE3:  Supported
    SSSE3:  Supported
    SSE4a:  Unsupported
    SSE41:  Unsupported
    SSE42:  Unsupported

Network Information:
    Network Speed:  

Operating System Version:
    Ubuntu 12.10 (64 bit)
    Kernel Name:  Linux
    Kernel Version:  3.7.1-030701-generic
    X Server vendor:  The X.Org Foundation
    X Server release:  11300000

Video Card:
    Driver:  Intel Open Source Technology Center Mesa DRI Intel(R) 965GM x86/MMX/SSE2

    Driver Version:  2.1 Mesa 9.0
    Desktop Color Depth: 24 bits per pixel
    Monitor Refresh Rate: 60 Hz
    VendorID:  0x8086
    DeviceID:  0x2a02
    Number of Monitors:  1
    Number of Logical Video Cards:  1
    Primary Display Resolution:  1920 x 1080
    Desktop Resolution: 1920 x 1080
    Primary Display Size: 18.78" x 10.55"  (21.54" diag)
                                            47.7cm x 26.8cm  (54.7cm diag)
    Primary VRAM Not Detected

Sound card:
    Audio device: 

Memory:
    RAM:  1994 Mb

Miscellaneous:
    UI Language:  English
    LANG:  en_US.UTF-8
    Microphone:  Not set
    Total Hard Disk Space Available:  22043 Mb
    Largest Free Hard Disk Block:  7254 Mb

Installed software:

Recent Failure Reports:

Terminal output from when starting the game in full-screen:

ExecCommandLine: "/home/mrshankly/.local/share/Steam/ubuntu12_32/steam steam://run/70"
ExecSteamURL: "steam://run/70"
Generating new string page texture 66: 128x256, total string texture memory is 442.37 KB
Generating new string page texture 67: 32x256, total string texture memory is 475.14 KB
CAPIJobRequestUserStats - Server response failed 2
System startup time: 20.41 seconds
Failed to load installscript /home/mrshankly/.local/share/Steam/SteamApps/common/Half-Life/installscript.vdf
ExecCommandLine: "/home/mrshankly/.steam/root/ubuntu12_32/steam steam://open/driverhelperready"
ExecSteamURL: "steam://open/driverhelperready"
Generating new string page texture 70: 64x256, total string texture memory is 540.67 KB
Game update: AppID 70 "Half-Life (Beta)", ProcID 2891, IP 0.0.0.0:0
ERROR: ld.so: object 'gameoverlayrenderer.so' from LD_PRELOAD cannot be preloaded: ignored.
Steam: An X Error occurred
X Error of failed request:  BadWindow (invalid Window parameter)
Major opcode of failed request:  20 (X_GetProperty)
Resource id in failed request:  0x2000075
Serial number of failed request:  5727
Steam: An X Error occurred
X Error of failed request:  BadWindow (invalid Window parameter)
Major opcode of failed request:  20 (X_GetProperty)
Resource id in failed request:  0x2000075
Serial number of failed request:  5728
Steam: An X Error occurred
X Error of failed request:  BadWindow (invalid Window parameter)
Major opcode of failed request:  20 (X_GetProperty)
Resource id in failed request:  0x2000075
Serial number of failed request:  5730
Using breakpad crash handler
ERROR: ld.so: object 'gameoverlayrenderer.so' from LD_PRELOAD cannot be preloaded: ignored.
ERROR: ld.so: object 'gameoverlayrenderer.so' from LD_PRELOAD cannot be preloaded: ignored.
Setting breakpad minidump AppID = 70
Forcing breakpad minidump interfaces to load
Looking up breakpad interfaces from steamclient
Calling BreakpadMiniDumpSystemInit
Looking up breakpad interfaces from steamclient
Calling BreakpadMiniDumpSystemInit
Steam_SetMinidumpSteamID:  Caching Steam ID:  76561198015223046 [API loaded yes]
Steam_SetMinidumpSteamID:  Setting Steam ID:  76561198015223046
saving roaming config store to 'sharedconfig.vdf'
roaming config store 2 saved successfully
Error initializing MSAA frame buffer
Missing shutdown function for Sys_InitLauncherInterface() : Sys_ShutdownLauncherInterface()

Missing shutdown function for Sys_InitMemory() : Sys_ShutdownMemory()

Missing shutdown function for Sys_Init() : Sys_Shutdown()

The specified video mode is not supported.
The game will now exit.
Installing breakpad exception handler for appid(gameoverlayui)/version(20130124161649_client)
Installing breakpad exception handler for appid(gameoverlayui)/version(1.0_client)
Installing breakpad exception handler for appid(gameoverlayui)/version(1.0_client)
Game removed: AppID 70 "Half-Life (Beta)", ProcID 2892 
Installing breakpad exception handler for appid(gameoverlayui)/version(1.0_client)
[0125/104138:WARNING:proxy_service.cc(646)] PAC support disabled because there is no system implementation
saving roaming config store to 'sharedconfig.vdf'
roaming config store 2 saved successfully
Installing breakpad exception handler for appid(steam)/version(1359078420_client)
Installing breakpad exception handler for appid(steam)/version(1359078420_client)