In theory, cooling could stop combustion if you could cool it with a very low temperature fluid, but I've seen nitrate film fully submerged in 5 gallons of water continue to burn happily. I believe that is the source of the idea that if you catch it in the first few seconds you may be able to stop it and that point there may not be that much heat, and you may have a chance to cool it. Once its going, it will be very difficult because the combustion process releases oxygen so you can't smother the fire.
As an interesting aside, in the late 1800s pool/billiards became very popular, and they started making billiard balls out of nitrocellulose instead of ivory. This caused occasional small explosions when the balls contacted each other.
I've never projected nitrate film, but I have projected old acetate film, and it was VERY fragile. I don't know if it was age or just the normal state of acetate film. Newer films were printed on what I believe was a mylar base, and that was practically indestructible.