Why do you observe an exponential decay ?
Because this is the behavior of the lifetime of the excited states. Radioactive decay appears in an exponential behavior in time.
Why is the number of events/counts for detector combination 1-2 smaller than for 1-3 ?
Because the nuclei detector 1 sees are oriented parallel to this detector. Detector 2 is oriented perpendicular to the direction of detector 1 and the direction of the nuclei so this means it is less likely for a photon to be emitted in this direction. Detector 3 is in the same direction as the orientation of the nuclei so it is way more likely also to see an emitted photon in this direction.