The videos after the sierra logo are distorted, as seen in the screenshot.
I've tried: changing compatibility mode, installing the community codec pack, nothing helps.
The .bik files play fine in the Bink video player and in VLC Player.
Weird thing: the sierra logo is also a .bik file in the movies folder... but that one plays fine.

Windows 10 / Geforce GTX 770

Anyone else had this? Solved it?

For me it works but bugged... It displays on the half of the screen... Strange. Possibly that it depends on installed video codecs.

You can ignore this problem, as intro videos don't affect game itself. Everything else should work fine.

Still haven't found a solution, and GOG support won't help because "Windows 10 is not supported".
I installed it in VMLite XP Mode, and there the videos look fine - so I can't imagine that it's an installed codec issue...

You launched it on a virtual machine without any additional codecs installed. That's why it possibly worked.
You could try to remove codecs from your main system to see if it works, but is it worth your time?

In my unofficial patch for Empire Earth 2, I disabled intro videos by default, because people experienced a lot of problems with them, and sometimes even game crashes. Except that, the whole game works just fine.

So I would recommend to skip it, as Gog nor anybody else won't fix it.