I understand that a lot of people have the issue dealing with the 3 second black screen, and then the game exits like nothing ever loaded. I had the same issue, and after some digging around in the dumps, I managed to fix it... and here's how:

1. After checking the dump, I found out that both d3dx9_24.dll and d3dx9_25.dll needed to be in the base folder ("..\Battlefield 2\"). You can find them at dll-files.com.
   
2. Edit your "My Documents\Battlefield 2\Profiles\Default\video.con" file with your favorite text editor (notepad here) and locate the 'VideoSettings.setResolution' and change the resolution to something like '1024x768@60Hz'. With the unsupported video cards, you will need to put a reasonable resolution in place of the 1024x768. You will not really be able to change it in your in-game video settings, only here. Also, make sure it says 60Hz. For some reason, the game doesn't understand the setting if it's set to 75Hz (thinks it's an invalid dynamic variable or something). I haven't tried other refresh rates, so you can play with it if you like.
   It should read:
   
   Code:

   VideoSettings.setResolution 1024x768@60Hz

3. Do that with any other profiles that you play with (under "My Documents\Battlefield 2\Profiles\")

That's all I had to do to get it to run on my unsupported video card. If in the event this doesn't work for you, your best bet would be to download the "Debugging Tool for Windows" (can be found at "microsoft.com/whdc/devtools/debugging/installx86.mspx"), install it, then go to where it installed and run "windbg.exe". Click 'File > Open Crash Dump', then open the .dmp files located in your "My Documents\Battlefield 2\dmp\" folder. It may give you a pop-up message, saying something like "Save information for workspace." Just hit NO. It should look something like:

Code:

****** (R) Windows Debugger Version 6.8.0004.0 X86
Copyright (c) ******** Corporation. All rights reserved.


Loading Dump File [C:\Documents and Settings\(YOUR USERNAME)\My Documents\Battlefield 2\dmp\bf2_v1.0.2442.0_2007-02-24_01.49.42.dmp]
User Mini Dump File: Only registers, stack and portions of memory are available

Comment: 'Battlefield2 MiniDump. Address: 00614C51 (In Windbg type: .ecxr)
ASSERT INFORMATION:
[...]
c:\dice\projects\bf2\Code\BF2\Game\GameServer\GameServer.cpp(3467): Error: GameServer: failed to initialize gamemode
[EOF]'
Symbol search path is: *** Invalid ***
****************************************************************************
* Symbol loading may be unreliable without a symbol search path.           *
* Use .symfix to have the debugger choose a symbol path.                   *
* After setting your symbol path, use .reload to refresh symbol locations. *
****************************************************************************
Executable search path is: 
Windows XP Version 2600 (Service Pack 2) UP Free x86 compatible
Product: WinNt, suite: SingleUserTS
Debug session time: Sat Feb 24 01:49:42.000 2007 (GMT-8)
System Uptime: not available
Process Uptime: 0 days 0:01:07.000
.............................................................................
This dump file has an exception of interest stored in it.
The stored exception information can be accessed via .ecxr.
eax=00000006 ebx=00000000 ecx=0012f864 edx=f73a0004 esi=00000478 edi=00000000
eip=7c90eb94 esp=0012f3ac ebp=0012f410 iopl=0         nv up ei pl zr na pe nc
cs=001b  ss=0023  ds=0023  es=0023  fs=003b  gs=0000             efl=00200246
*** ERROR: Symbol file could not be found.  Defaulted to export symbols for ntdll.dll - 
ntdll!KiFastSystemCallRet:
7c90eb94 c3              ret

Obviously, your error may not be the same... this is just an error I had a while ago.

This should have some information that will be useful to help you find out what the issue is.

NOTE: The two DLL files at the top may not be the same ones for all users. I encourage you to open the .dmp files to find out what exactly is crashing your game.