I've noticed that Starfighter is not playing the FMV files. It seems to be skipping them and going directly to the mission. I've tried both available configuration options "high quality cutscenes" enabled/disabled, but nothing works. Is there a way to fix this? I'm running Windows 7 64-bit.

Are you playing on a laptop? Because on most laptops now, you get an Optimus GPU system, which might have issues with some games (right-click on your game shortcut and see if you have an option saying something like 'launch with graphic card', and you should have two choices, Intel HD and nVidia).
If you do have Optimus, try window mode (just seen you have tried this already in your other post), which should be the best option performance and bug-wise, or launch the game with IntelHD (it works well with that, it seems). On my laptop, these solutions allow for videos to be played.

Or try installing a codec pack - K-Lite is the one I'm using, I think another well-known one is CCCP (?).

I'm using a regular computer with an nVidia video card. I'm using Win7. I've been running Starfighter in a window as well. Do you know what codec is needed for the .bik video files?

.bik files are read with Bink Video, you can download the RAD Video Tools here: http://www.radgametools.com/bnkdown.htm . This should at least allow for videos to be played outside the game.

Is there anyway to fix this to allow the game to play the video files on its own?

Maybe installing a codec pack rather than just Bink, particularly if playing windowed doesn't make a difference.

However, Bink videos, as far as I know, get everything they need to play from some .dll files delivered with the game. But on some games they tend to have compatibility issues, and in this case using older Bink .dll files may help.

There's a thread for another game discussing a different issue with videos, but they are compressed with Bink too, so this might help: https://www.gog.com/forum/vampire_the_masquerade_redemption/jittery_videos
Pictures linked as jpg in this thread are actually renamed bink32.dll files, just right-click and 'save the target element' and rename your .jpg file as bink32.dll. Maybe changing the game's .dll file to one of these will help.

If you already have a codec pack installed, try blacklisting the game in the pack settings to see if it's not messing with it (I've got the latest K-Lite and had no issues though).