After 2 hours of scouring the internet, I found the latest, manufacturer-specific driver and installed it, but no dice.  The only thing that has changed is the date in my debug report.

Debug Message:  Version v1.80e Dark Avatar last updated on: Thu Nov  8 10:48:23 2007
Debug Message:  Checking DX Version.
Debug Message:  *********DXDiag info follows.*********
System Info
Time: 12/16/2010, 10:57:38
DirectX Version: DirectX 9.0c (4.09.0000.0904)
Machine Name: BL-RH-YOUNGAJ
Operating System: Windows XP Home Edition (5.1, Build 2600) Service Pack 3 (2600.xpsp_sp3_gdr.100427-1636)
Languages: English (Regional Setting: English)
System Manufacturer: Hewlett-Packard
System Model: HP Compaq dc7600 Small Form Factor
BIOS: Default System BIOS
Processor: Intel(R) Pentium(R) 4 CPU 3.20GHz (2 CPUs)
Physical Memory: 3320MB RAM
Page File: 408MB used, 4260MB available
Windows Dir: C:\WINDOWS


Display Info
Display device 1:
Device Name: \\.\DISPLAY1
Card Name: Intel(R) 82945G Express Chipset Family
Manufacturer: Intel Corporation
Chip Type: Intel(R) 82945G Express Chipset Family
DAC Type: Internal
Key Device: Enum\PCI\VEN_8086&DEV_2772&SUBSYS_3011103C&REV_02
Display Memory: n/a
Current Display Mode: 1024 x 768 (32 bit) (60Hz)
Monitor Name: Plug and Play Monitor
Monitor Max Resolution: 1600,1200
Driver Name: igxprd32.dll
Driver Version: 6.14.0010.4864
Driver Date: 8/24/2007 11:22:52
DirectX Acceleration Enabled: Yes
 


Sound info
Sound device 1:
Description: Realtek HD Audio output
Default Sound Playback
Default Voice Playback
Hardware ID: HDAUDIO\FUNC_01&VEN_10EC&DEV_0260&SUBSYS_103C3011&REV_1004
Manufacturer ID: 1
Product ID: 100
Driver Name: RtkHDAud.sys
Provider: Realtek Semiconductor Corp.
Driver Version: 5.10.0000.5645
Driver Date: 6/17/2008 15:49:22
Type: WDM
Other Drivers:
Acceleration Level: 3
Cap Flags: 0
Min/Max Sample Rate: 0, 0
 


Debug Message:  *********END dxdiag info.*********
Debug Message:  Entering WinMain
Debug Message:  ******* Starting Game Shell *******
Debug Message:  Preparing to change display settings.
Debug Message:  Failed to find a device that can start with hw vp mode. Trying mixed vp.
Debug Message:  Failed to find a device that can start with mixed vp mode.  Trying sw vp.
Debug Message:  No compatible devices found.
Debug Message:  InitializeDirect3D Failed.
Debug Message:  Now windowed.
Debug Message:  Shell initialization complete
Debug Message:  GalCiv2 is activated.
Debug Message:  WM_CLOSE message received.
Debug Message:  GameUninitialize start

Nope.  Is there any other useful information in my debug report?  If not, then it must be a deeper problem with my computer, because it doesn't run any of my other old games, either.

Thanks anyway for your help, Kryo, I really appreciate it.  Maybe in a year or two I'll get a new videocard or a whole new computer, and then you can bet that the first thing I do is fire up Galactic Civilizations II (and maybe even spring for the Twilight of the Arnor expansion).

Success!  It came to me in a dream, in which I turned on my computer and Dark Avatar started miraculously.

So, this morning, after determining that it was in fact a dream and the game still wasn't working, I was filled with the vigorous spirit of repair.  So, I went through a plethora of methods, including disabling my driver entirely.  Finally, I stumbled onto the Troubleshoot tab on the Advanced tab of the Display settings.  The hardware acceleration was set to None (as recommended for computers that are having graphics issues.  However, sliding it around I discovered that one level disabled all DirectDraw and Direct3D accelerations.  I turned it up one past that level, and restarted my computer.  And, upon starting up the game, it worked!  It goes VERY slowly, but with all of the graphics turned down, I believe it will work.

Thanks for all of your help, Kryo, and I hope this helps other people with problems, too.