At one point, the voice in Cloud's head tells him, 'It's started moving..." referring, I believe, to Jenova, who was just about to break out of Shinra's confinement at that time.
I believe that Jenova may have broken herself out of the tank - after all, look at the damage to the building. She then may have needed Sephiroth's guidance in order to navigate the outside world. She did have no head, which would make it somewhat difficult. So Sephiroth (or his manifestation, if you believe that theory) picked her up and carried her off after nailing President Shinra to the floor with his Masamune.
As for the timing of Jenova's escape/ release? It may have something to do with the Reunion. There were many members of SOLDIER in the Shinra building. Cloud, also, had been drawn there by fate - or was it a more powerful compulsion? Anyway, a lot of carriers of Jenova's cells were gathered in the one place. That may have given her the strength she needed to break out, in order to head for the North Cave to complete the Reunion.
Now, about Cloud's door:
If the door was opened so that Cloud could escape and free Jenova, then Tifa would have heard the considerable ruckus going on outside. Don't forget that Jenova's rampage tore huge sheets of metal walls, which makes a fair racket.
It's only fair to assume that the door was opened after Jenova had left.
Notice how the dead guard isn't surrounded by that red goo, and there's no heavy structural damage to the walls near him? This suggests, to me, that he (barely) survived Jenova's onslaught, and later crawled to the only people left alive on the floor - Cloud and co. He needed help. He'd have been too badly injured to think of, or implement a better plan; he couldn't take the stairs because that's the way Jenova's trail led. Going to the elevator would've required going a long way, past innumerable escaped monsters. So I figure that he staggered back to his post - the cell wing - and unlocked Cloud's door before stumbling to his death at the end of the corridor. By this reasoning, the door wasn't opened until after Jenova had left. Cloud and Tifa wouldn't have been woken by the noise of the fighting.(I know the team could easily whisper from one cell to the next, but it's fair to assume that the cells would have their own ventilation system, which would easily conduct sound from one room to the next, but not to the corridors outside).
Just another strange theory from me, but it fits <most of> the facts fairly well, without suggesting any deep conspiracies involving Hojo's deception, Cloud's mindlessness, or the unspeakable vengeance of Sephiroth's long-lost assistant under-secretary.
*Waits for Phoenix 17 and Raistlin to spot the holes, which are probably perfectly obvious except I've somehow missed them*
They come to the conclusion that they should be a god. -_- -_-
