a) I think this exponential decay is linked to the decay-process of the intermediate state. Close to the lifetime, there’s a high chance that the nucleus will decay. It’s less probable that a nucleus will only decay much later than this lifetime. Therefore, more nuclei in this intermediate state will decay around this lifetime, and less much later; this in an exponential way.
b) This has to do with the probability distribution of the emitted radiation. It’s more probable that the gamma-quantum is emitted in the direction of the nuclear spin, and this lies here along the axis of detector 1 and 3. The direction of detector 2 lies along a direction in which the probability is lower, and thus detector 3 will have higher probability to detect more gamma-rays than detector 2.