I went to Microsoft to get the xp compatibility mode download, but it didn't give me a validation code. The dialog box was supposed to have an "Open" button, but it didn't. it only had "Save" and "Cancel"...so, I saved, then went to Downloads, which is where the pathway said it was going, but it isn't there. I also went to Documents, but it isn't there, either.
Were you download
Windows XP Mode? This paragraph seemed a little unusual to me, as it shouldn't be necessary to download anything to set XP Compatibility. Windows XP Mode is another option that some programs need (if they really only work in XP), but Civ3 should run in Windows 7, albeit not always as smoothly. As I remember, XP Mode also only runs on Windows 7 Professional, which might be why you didn't get a validation code (if you are running Home).
Vuldacon's post #11 is probably the right way to set regular XP compatibility mode in Steam, although I'm not very familiar with setting compatibility in Steam.
For non-Steam, the way I usually go about it is right-clicking on the program (or a shortcut to it), and choosing Properties. Then, I go to the Compatibility tab, where you can set the compatibility mode:
Either XP or 98/ME might help, as Civ3 supports both well.
I'm still curious if the Civ3 and/or Steam Civ3 installs are somewhere within Program Files. Even with that, the errors such as Bink32.dll not being available are rather strange. There should be a copy of that file both within the main Civ3 install folder, and within the Conquests folder inside of the main Civ3 install folder, so that it isn't being found is odd.
It's also worth noting that in general, compatibility issues with Vista also apply to Windows 7 and Windows 8. Windows 7 is essentially an improved version of Vista, and the causes of most of the compatibility issues that older programs had with Vista are still present in 7 as well. The official version number of Vista is 6.0, whereas Windows 7 is version 6.1 - it's an improved version of Windows Vista, but not drastically different at the core, just like XP is an improved version of 2000, and 98 is an improved version of 95. So, particularly for Civ3, it's not unusual to see compatibility posts refer to Vista with the meaning implicitly being "Vista or later" (although I have seen some posts about Windows 8-specific issues).
It is normal to see the Civ3 Complete screen when starting from Steam, however.
I think it might be helpful if you could take a screenshot of your CivIII install folder and post it. That would both highlight any missing files, and clarify the install folder situation. Mine looks like this (note there are a couple files there that I've added, so you likely won't have everything I do):
One of the key factors in the above (if I were running Vista/7/8 instead of XP) being the install being to D:\, and not C:\Program Files. In my case, I just have it there since I have more space on D:\ and it's shorter to type, but if I have Vista/7, it would be much more important.