The Door to Deepground

That door that the rescue team is opening on TV at the start... do we see it at any point in the rest of the game?

No we do not. It has been a long time since I played DOC, but IIRC Vincent gets down under the Shinra Building through a different way.

I'm 99% sure Tifa is literally telling him where to go at that point in the game and it's explained that there was another entrance.
I could be wrong, it's been a while.

Like Sicarius said, the door to Deepground in the FMV is not the same as the one you enter in-game in Chapter 9. You can see for yourself in this longplay video and then compare it to the intro FMV. Not even the corridors Vincent run through, starting at 49:17 in the longplay video, look anything like the corridor that the team in the FMV is running through at 4:21.

As I thought. Figured I'd check with more observant people than me.

And I just noticed Vincent accessorises his phone with a Cerberus sticker.

In gameplay dialogue, Shelke mentions that Vincent took 'the President's entrance.'