1-a) Exponential decay is typical for nuclear decays. Say you start with 100 nuclei that have a certain probability of decaying within a given time interval. For example 10 percent in 1 sec. After 1 second realistically about 10 will have decayed (so we would see 10 signals in best case scenario) and we are left with 90 nuclei that can decay. If again 10 percent decays we only see 9 signals. Then 8,1 signals, then…
It is possible to mathematically prove this gives an exponential function, but this reasoning should suffice.
1-b) This is due to the anisotropy. We see in the picture that in general decays are more likely to happen to the direction of detectors 1 and 3. And less likely to direction of detector 2. So of course the chance of detecting a signal in detector 3 (as a follow up of detector 1) is higher than detecting one in detector 2 (as a follow up of detector 1). This gives a higher count in 1-3 than in 1-2.