Time paradoxes are the devil's snare for underemployed minds. They're fun to consider in a 'what if?' sort of way. Film makers and authors have dealt with this time and again in a host of films and television including 'Star Trek: First Contact', the 'Back to the Future' trilogy, 'Bill and Ted's Excellent Adventure', 'Groundhog Day' and the Stargate SG1 homage, 'Window of Opportunity'. Heinlein's 'All You Zombies' was written decades ago and yet it will still spin out people reading that short story for the first time.
In the case of Terry Gilliam's excellent film, '12 Monkeys', it's hard to establish what may be continuity problems versus plot elements intended to make us re-think our conception of the film. Repeated viewings will drive us to different conclusions if we retain an open mind.
Some, seeing the film for the first time, will regard Cole, played by Bruce Willis, as a schizophrenic. Most will see Cole as a man disturbed by what Adams describes as 'the continual wrenching of experience' visited upon him by time travel.
Unlike other time travel stories, '12 Monkeys' is unclear as to whether future history can be changed by manipulating events in the past. Cole tells his psychiatrist, Railly (Madeleine Stowe), that time cannot be changed, but a phone call he makes from the airport is intercepted by scientists AFTER he has been sent back to 1996, in his own personal time-line.
Even this could be construed as an event that had to happen in a single time-line universe, in order to ensure that the time-line is not altered...Cole has to die before the eyes of his younger self for fate to be realized. If that's the case, time is like a fluid, it always finds its own level or path, irrespective of the external forces working on it. It boggles the mind to dwell on this sort of thing too much.
If you can change future events that then guide the actions of those with the power to send people back in time, as we see on board the plane at the end of the film, then that means the future CAN be changed by manipulating past events...or does it? The film has probably led to plenty of drunken brawls at bars frequented by physicists and mathematicians