Collision detection is only available during the Move or Rotate Component command. As you mentioned you can use physical dynamics, but it relies on the existing mates to define a part's degrees of freedom. Like them or not, mates are necessary.
As a note, for collision detection to work, the parts cannot be touching in any way to begin with. So all parts would have to be mated and designed with some space between them and the other parts. I don't think any of them could even be flush.
I did collision detection between two gears, to simulate the movement of one gear based on another gear's rotation. When the tooth of one gear hit a tooth of another gear, that second gear was spun accordingly.
Even this simple example nearly brought the computer to its knees. I don't believe simulating an entire bike chain this way would work very well.