If James Cole (J.C.) is insane, then:
How did he escape from the locked room?
How did he know the boy was not down the well but hiding in a barn and using a walkie-talkie?
You are definetly supposed to question if he is sane and from the future. The on screen text that comes at the beginning was added test screenings and the TG felt the audience was accepting the ‘sane JC’ too easily.
If Cole is insane then he also has precognitive powers. His delusions are accurate visions of the future. For instance, early in the movie he sees zoo animals wandering around loose. But they don’t actually get released until the end.
Simpler to believe that he’s really a time traveller.
Besides which, how did Jeffrey survive the plague? Much less find a way, on his own, to travel back in time?
Early in the movie, he also has a flashback where he “saw” Jeffrey Goines as the man in the airport. So it’s clear that his memories of events are mutable.
The scene where Cole is given the gun threw me off a little…and I came up with an alternate theory. My theory is that the scientists, knowing that Cole couldn’t finish his primary mission (getting an original strain of the virus) since he destroyed his tooth, needed to get him to somehow point out the virus-carrier in an unambiguous fashion to the next time traveler. So when the next volunteer goes through, they’ll be able to witness the airport incident and say “hey, Cole just tried to kill THAT guy, so he’s the man”.
Since it’s now impossible for the scientists to bring Cole back, he can’t describe the guy to a sketch artist or anything like that either, so they came up with the gun plan…I dunno…not sure if that theory holds up better or worse than any other, but it’s an alternative to the theory that the scientists changed their mind about whether or not history could be changed.
My version of 12 monkeys has a very nice long documentary on it (almost as long as the actual movie). The final scene with Jones and the virus carrying assistant on the plane was debated a bit before including it in the movie. Terry Gilliam didn’t want to include it, but worried audiences wouldn’t “get” the ending without it, so a decision was made to include it. Jones did NOT stop the virus from being released and spread. The mission of James Cole and the scientists is to secure the virus so they can develop a vaccine for it. Remember, they all lived underground still because of the virus. They wanted to be able to immunize themselves against the virus and move back above ground, not prevent the virus from being spread in the first place (which probably would have been impossible).
As for the question posed by the OP, I suppose it’s possible, but I don’t think there’s really enough foreshadowing for it to be true. It would require Jeffery Goines to live through the catastrophe, somehow work out a way to travel back in time before the scientists did, then work in the past alongside his own dad for at least a year or two and develop the virus. Sounds kind of far-fetched to me. On the other hand, I could see Terry Gilliam putting something like that in the movie, knowing that only a few people would notice it.
Other people have posted more negative theories. One is that the scientist wasn’t there to ensure the virus sample was obtained - she was there to ensure it was released. If Cole had succeeded and stopped the virus from being released, then the world would not have been depopulated - and the science council wouldn’t be in charge. The scientists might have decided that the pros outweighed the cons overall (and they could quiet their consciences by telling themselves they were just making sure that history wasn’t changed).