Here is an article of how Russia's nuclear arsenal is controlled
That's basically identical to how it works for the US as well. The Nuclear Football contains the launch codes, retaliation options and some sort of communications ability. But all that really does is enable the US President to authorise the release of nuclear weapons. The launch orders are generated and actioned by the US military. It would seem that Russia operates on the similar basis with the Cheget (their Nuclear Football) being used to allow the President of Russia to authorise the release of nuclear weapons with the actual commands transmitted by the Russian military.
Submarines listen to Radio 4! If it goes silent then they authority to open the PM's letter on what to do next.
Er not quite. The UK ballistic missile submarine on patrol uses a variety of methods to keep an eye on whether or not a catastrophe has occurred. For one thing they monitor regular message traffic (one way, they listen, they don't transmit) from the MoD, if that were to go down for a period of time then they would move to alternatives such as sampling the atmosphere (is it, you know, radioactive) and also listen out for civilian transmissions, such as Radio 4 Long Wave (preferred as long wave of course has a tremendous range). It's not quite as simple as there being a technical fault at Broadcasting House triggered a UK nuclear response!
Bush opened the Football during 9/11 only to realise that the 'Where to bomb' book was about 30 years out of date so it had to be updated. Codes where not entered, so they say.
Are you sure about that? I'm not aware of any sources that state that Bush had to open the Nuclear Football, why would he, it wasn't clear who was attacking the United States? There's even less to suppose that any launch codes were being bandied about. Again, why would they? Who were they going to use nuclear weapons against? Sounds rather like hyperbole to me...
My understanding is that Russian and US airmen directly engaged and shot each other down 18 times between 1950 and 1970; how many nuclear wars did that cause?
None, thankfully, as cooler heads prevailed. But that isn't the risk. The risk isn't that US/NATO and Russian aircraft shoot each other down and Russia retaliates with a nuclear strike. Why would they? That would be insane as they'd be killed by NATOs second strike. And whilst I have my doubts about Putin I'm fairly certain that he isn't a) insane and b) wants to rule over more than nuclear cinders.
No, the risk is miscalculation. It's nuclear war by accident when no-one intended to start one. It's the miscalculation during a crisis when everyone is under incredible stress, when everyone is tired, when everyone is scared. That's when a nuclear war will start. Not through some cold blooded and calculated first strike but when someone screws up catastrophically.
We've nearly been there. During the Cuban Missile crisis the biggest danger wasn't that Khrushchev or Kennedy would actually use nuclear weapons it was when a Soviet submarine commander, who had been out of contact with Moscow for several days, felt that he was under attack by the US Navy (who were dropping training depth charges to try and get him to surface but which could cause no real damage) and considering the grim nature of the last news he had received attempted to use a nuclear torpedo against the 'attacking' Americans. He was unable to do as as it required three officers, himself, the Political Officer and, as they were aboard, the submarine flotilla commander to all agree. The flotilla commander did not. The submarine surfaced and there was no war. This all went down at the climax of the crisis. If group of US Navy destroyers had suddenly been vaporised by a nuclear strike, that no-one in the Soviet Union had authorised, the results would surely have been catastrophic.
Or there was the incident in 1979 where training tapes were accidently fed into the live NORAD computer system which simulated a massive Soviet first strike on the United States. This occurred during a time of comparative stability and within a few minutes satellites had been able to confirm that there were no Soviet missiles inbound. But if it had occurred during a crisis of the tension of Cuba? If it occurred whilst NATO and Russian forces were engaged active combat? Or the famous 1983 Petrov incident where Soviet satellites reported US launches due to sunlight reflecting off of clouds over the mid-western United States but for one man deciding it must be a false alarm and failed to report it up the chain of command. Do we have confidence that if NATO forces were bombing Russian troops in eastern Ukraine and a report is made by Russian early warning systems that a NATO nuclear strike is underway that a modern day Petrov wouldn't pass that up the chain of command? That given minutes (NATO submarine launched ballistic missiles are probably no more than five or ten minutes flight time from Moscow depending on deployment zone) to decide if this is real or not that Putin will not order a counter-strike?
I agree, a NATO fighter shooting down a Russian aircraft will not start a nuclear war. I also don't think that Putin is crazy enough to start one (though maybe Boris is if he needs to distract from another domestic political issue though

). I do however think that in the heat of a crisis it is highly probable that even a tiny miscalculation could start a devastating nuclear war that no-one actually wanted or intended. That is the real risk.