1-a) I think the reason for the exponential decay in the given graphs has to do with the fact that each of the decays (gamma-1, -2) is already described by an exponential decay. The number of detector events in function of the time difference of both decays should then also give an exponential decay.
1-b) This has to do with the accompanying radiation probability. The probability for an event to be detected in detector 1 is about the same as in detector 3. This can also be seen from the radiation probability form. For detector 2 it is the opposite case, here the probability is lower than for detector 1 and 3.