At first, I had big problems with messed-up menus. However, with help from Support (thank you Andrew) and after trying a clean install of Steam and then CoD2, I finally got it to download and install properly with the right menus.
I had a problem with the graphics: everything looked hazy, but this seems to be a problem with a few of my games so it may be an issue with my computer.
To solve it, I went to the bottle for Steam and ran regedit.
I then went to HKEY_Current_User/Software/Wine and created a Key called Direct3d. I then created a String inside that called useGLSL and gave it the value enabled (right click and enter without quotes "enabled" in the value field.
No problems loading, running, etc. All looks perfect!
Thanks to the codeweavers staff and Andrew in particular for getting CoD2 running!