When temperature is decreased, why do reactions occur at all? I admit that my knowledge of collision theory may be lacking, but, as I understand it, when particles collide, a reaction will not occur without overcoming the activation energy. 

That being said, as the temperature of the environment in which the collision takes place is decreased, I believe it is logical that the kinetic energy of these particles will also decrease. Hence, to me, it would make sense if none of these particles had the energy required to react.

So, my question is: Why do reactions still occur when the environment's temperature around a collision decreases? That is, shouldn't there be a point (such as the temperature in a freezer, perhaps?) in which the activation energy cannot be overcome at all?
The activation energy varies a lot for various reactions. As long as T > 0 K, there is still disordered kinetic energy in the molecules. But the reason gasoline doesn't spontaneously combust, is that the temperature isn't high enough to overcome the activation energy - a spark is needed. So you are onto something, just remember that the activation energy varies with each reaction.