If your interested, get the book 'EMERGENCY Crisis on the Flight Deck' by Stanly Stewert. It is a fantstic book full of stories (about 10) like that. It also includes the 'Gimli Glider'. The book goes into a LOT more detail than the film and is also more factual, giving the full reason why the fuel ran out. It is quite a complicated set of misfortunes and misunderstandings that caused this to happen.
Anyway, your questions.
The pressurization system would indeed fail if the engines ran down and their would be a limited oxygen supply available, your right. However the pilots had already started a decent into winnepeig as they were aware of a problem. They quickly decend below 10,000ft were the pressurisation system is no longer needed for comfortable breathing oxygen. In other words, the pressurization system DID fail. It just did not matter because they were low enough.
The generator is called an RAT, or Ram Air Turbine. It provides hydraulic power to the flight controls and the instuments on the flight deck run off of a battery. The EFIS screens (those big screens in front of the pilots) in real life would go blank. The film is unrealistic here. ATC would theirfore be able to pick up a trace and did in real life. ATC never lost the aircraft. The RAT deploys when N2 (the high pressure stage of the engine) drops below a certain value, so yes, the whole 'nick of time' thing was down to hollywood.
I hope this helps. I cannot answer your last question, I am not a pilot. (yet anyway!)
BTW: Did you notice that in parts of the film that F/O Quintail had one F/O's eupelete and one captains one?
Cheers
Richard
Edited for spelling