The Cutscene that opens the game seems to play fine. Does this mean that I don't have to worry about having the cutscene will not play problem that some people are having?

If the first one plays, they all do. There's only like 7 through the entire game:

Intro
Military Branch selection
year 1
year 2
year 3
waking up on von braun
and the ending.

So even if they don't work the only two you'd be missing are the intro and ending since the others don't actually need to be there.

In my game, the intro plays when I click the INTRO button but the training videos don't.

''Waking up on Von braun'' cutscene didn't play for me, but the Intro did. I guess it's a weird random thing whether a scene plays out or not.

try the last resort fix from the sticky.

Is it possible that GOG scrounged the cutscenes together from different sources? When was the last time they were reencoded?