If it has not already been suggested try downloading / installing
DirectX 9.0c End-User Runtimes (June 2010).
Some older games cannot find the the required DirectX 9 dll's on a Win 7 box. Installing the runtimes will fix the problem if it's the DirectX runtime files that are missing. I have had to do this on several Win 7 boxes.
Edit: I would uninstall the game prior to installing the runtime package. Be sure to install the runtime as an admin.
On the new install be sure to:
a) right click the setup.exe and click "Run as administrator" in the resulting dialog.
b) on the first screen of the GOG installer, click the "Options" button in the lower left corner. Point the install to a directory outside of Program Files. I use C:\Games but any directory name and location not in the Program Files directory will do. I would suggest creating the directory before you start the install.
c) After installing the game and before running it, right click the game shortcut, click Properties, click the Compatibility tab and checkmark "Run this program as an administrator" I also check "Disable visual themes" and "Disable desktop composition" as well.
Try starting the game . . .
If the game still won't run, your firewall may be the problem. Some firewalls with throw up a dialog to "Allow" a file to run. This may be the problem if, when you start the game, it just stops on a load screen or a black screen and loads no further.
Comodo in particular does this and sometimes an AV program will as well. Some games,
especially older games, will block the request from showing and the game will not run. ALT-Tabbing out of the game will sometimes let you see the requests. Click allow for each request and it should not ask you again. There may be several "allow" requests on first run so ALT-Tab out each time the loading stops. It is often easier to mark the game exe as a trusted program in your firewall prior to running the game.
A quick way to determine if it is the firewall or AV stopping the game is to disconnect from the internet and disable the firewall and AV programs. If the game runs, stop it and enable the AV program, if it runs, stop it and enable the firewall. Once you determine which program it is, make the games exe a trusted file in the blocking program. Enable both prior to reconnecting to the internet.
I have no further suggestions for you other than to maybe . . . give this community a chance to help you if you have a problem. Check the game forums for a solution and post your problem there. If all else fails open a help ticket with GOG support.