I got the submarine on disc 2, and when I tried going into Crystal Cave, there was some sort of invisible force blocking me in the tunnel. What could be the problem? Am I supposed to go there in disc 3?

As far as I know, there is no such force field in the tunnel leading to Lucrecia's cave - however, there is a sharp bend in the tunnel if I remember rightly, which unless you happen to turn the camera view will result in the sub getting stuck on the tunnel wall. Try shifting the camera angle with the shoulder buttons (on the PSX/PSN versions - I can't remember how to do it on the keyboard in the PC version).

Failing that, all I can think is it might be a bug, in which case I have no idea how to overcome it, other than breeding a mountain-crossing chocobo (green) and riding over the mountain to the cave.

I think it was a bug, because I can get in now. I heard you can get Vincent's final limit break here, but I don't know how.

You can, by heading into the cave and speaking with Lucrecia (this triggers a long cutscene revealling Vincent's past). However, even after the cutscene, you don't actually get the Limit Break til later on (like, in disc 3).

Just wait til later in the game before returning to the cave and you'll find an item sitting on the crystal altar where Lucrecia was previously. This has both Vincent's level 4 limit break and his ultimate weapon the Death Penalty in it. :) Make sure you have Vincent in the party though as the item won't be there if he's not in the party.

Now ya tell me. :\ I just now beat the game, didn't even know you could leave the Northern Cave on disc 3.