Imagine that we number the caterpillars from 1 to 100, and that each caterpillar starts out with a baton labeled with its number. Suppose that when two caterpillars meet, they swap batons before turning around.
Though a caterpillar turns around many times, its baton travels uniformly at 100 meters per hour, turning around each time it gets to an end of the wire.
Suppose two hours have elapsed. Each baton has traveled 200 meters and returned to its original position and direction of motion. Since the relative order of the caterpillars never changes and of the relative order of the batons is the same as it was when they started out, each caterpillar now carries its own baton again. Each caterpillar must be in its starting position and orientation.
Suppose only one hour has elapsed. The positions of the batons have been reflected about the center point of the wire. Since the caterpillars are placed randomly, the probability that a baton started at the exact center point or that two batons started at the exact same distance from the center point is 0. No baton is in the starting position of any baton, and since every caterpillar carries a baton, no caterpillar is in its starting position.
Solution to the addendum:
A caterpillar, Alice, has been added to the exact center of the wire. The analysis after two hours is the same as the above: Alice must be in her starting position.
After one hour, the positions of the batons have been reflected about the center point, and Alice's baton is at the center point. Alice is at her starting position if and only if she now carries her own baton.
Since the relative order of caterpillars never changes, this is the case if and only if she is the median caterpillar on the wire. This happens if and only if, out of the other 100 caterpillars, 50 were placed to the left of the center point and 50 were placed to the right. Since a caterpillar has a probability of 1/2 of being placed to the left of the center point, the probability of this is (1/2^100)(100!/((50!)(50!)).