I was thinking about how NATO and the Warsaw Pact both possessed extensive conventional forces in addition to their potent nuclear arsenals, and I thought about how separate those forces are (dedicated nuclear bomber wings and missiles as an example, both of which aren't necessarily forward deployed).
However, that brought up the question of what the SLBM carrying SSBNs on each side would do in the event of a conventional conflict. They were often deployed close to their opponents so that they could minimize response times, but they were also often tailed by so-called "hunter killer" submarines. Since what-ifs aren't in the spirit of this subreddit, I think a more useful question would be along the lines of what military planners on both sides thought should be done about these potent nuclear forces that were at the frontlines of potential conventional conflict. Were there any agreements between the superpowers over whether these submarines (which had significant stealth characteristics, powerful torpedoes, and long running time making them quite capable of conventional combat operations) would be attacked in a conventional conflict? Was there any worry of whether or not an SSBN under threat may attempt to launch their nuclear payload as a sore loser response? was there any training or measures to prevent this from happening?
Unfortunately, I don't know if it is possible to answer your question. The biggest issue is that your premise of a "conventional conflict" between the US and the USSR during the Cold War is itself a very glaring what-if.
This is because the entire US strategic playbook during the Cold War was solely focused on nuclear war and nuclear escalation. The reason for this was that the US very clearly recognized that the Soviets had a huge advantage in terms of conventional forces. US strategists were entirely pessimistic (or perhaps realistic?) about how a conventional conflict with the Soviets would play out. Hell, US strategic planners in 1947 predicted that the Russians would be able to control literally all of Europe within just 45 days of launching a invasion. Military projections for Russian capabilities in the 50s and 60s were even grimmer, given that the Soviets would have fleshed out their nuclear arsenal by then.
Because of this, the entire basis of the Cold War and US strategic thinking during that period was the use of nuclear weapons. Before the Soviets fully developed their arsenal in the 1950s and early 1960s, the US war plan was to counter the Russian conventional advantage with the US nuclear advantage, given that the Soviets at this time had not fully developed their nuclear capabilities. So the Soviet troops would be allowed to swarm across Europe, only to be bombed to oblivion by US nukes, which would hopefully be enough to push them back or at least create a stalemate.
After the Soviets gained nuclear parity with the US in the 1960s though, the US strategy changed slightly. Now instead of launching an all-out nuclear strike against the Russians, the President would have a list of retaliation options to chose from, so that he could hopefully choose how far to escalate the now equally nuclear-powerful Russians. Even at this stage though, EVERY SINGLE OPTION on this list was a nuclear one.
Conclusion (sorry it took a long time in coming, but I absolutely had to emphasize the US position to make my answer clear): If a conventional conflict ever broke out between the US and the USSR during the Cold War, the US strategy would be to go nuclear and hopefully stalemate the Russians before all of Western Europe was lost. In this scenario, the SSBNs would be used as intended, to rain nuclear hellfire upon all Russian strategic targets. Simply put, the US never even bothered considering conventional war against the Soviets; that was seen as suicide.
On the other hand, if by "conflict" you perhaps mean accident or something like that, then it's easier for me to answer your questions. In terms of SSBNs launching their nukes on their own initiative or as a last-ditch move, there were very many countermeasures in place. In both Soviet and US SSBN protocol, the launch of nuclear missiles required both a direct order from the outside world and the transmission of launch codes. Without these, the sub crews were basically blocked out of any sort of rogue launches. As for their training, it really was just to follow orders. If every single redundant launch protocol was fulfilled, then they were trained to fire away, no questions. If even one part of that protocol was off, then it was not supposed to be followed. Nuclear sub commanders (as well as missileers in silos) were given as close to 0% personal input as possible, and that was the intention.
Sources:
American War Plans, 1945-1950 by Steven Ross is what I referred to when talking about early US strategy for matching Soviet conventional might with the use of nukes.
You can also try looking up Cold War era US Single Integrated Operations Plans (SIOP) if you are further interested in learning about later US strategies for using coercive nuclear force to stalemate the USSR.