Great question!! If I understood well, for "frequent flyers" (FF) the time interval between visits in your hospital follows a exponential distribution with a short half-life parameter ($\theta$).
To me it looks like the visits of the non FF patients also follows an exponential distribution, but with a much larger $\theta$ (correct me if you have evidence against this). The most straightforward model would be to assume that there exists a cut-off value $\theta_0$, (e.g. 1 month) such that for a FF $\theta < \theta_0$. And we now need to set up a statistical framework to test this.
Let us assume for now that you manage to get a good estimate of $\theta_0$ (more on this at the end of the post). Like @Gschneider pointed out, the number of visits in a time interval $t$ will be Poisson. So the probability that a non FF visits your hospital $n$ times or more in a time $t$ is lower than
$$\sum_{k=n}^{\infty} e^{-t/\theta_0}\frac{(t/\theta_0)^k}{k!}.$$
You can compute this score for every patient, $t$ is the time from first visit, and $n$ is the number of visits minus 1 (do not count the first visit). If it is below 0.05 you can reject the null hypothesis with a risk less than 0.05 and conclude that they are FF.
Now your point 2) is more difficult. Before embarking on advanced methods like the scan statistics, 2-state Hidden Markov Models, or breakpoint detection, I would suggest to keep it simple as a first approach. If the model above gives good results, you could decide on a maximum number of visits you are ready to accept before taking action. Say 3 for the sake of the example.
You can estimate the steady-state $\theta$ of a patient by the mean time between visit, excluding the last two visits. With the oversimplistic assumption that this estimate is perfect, the sum of the last two times between visits, say $t_0$, is distributed as a $\Gamma$ variable with parameters $(2, \theta)$. You can use software like R to compute the value of
$$\int_0^{t_0}\frac{1}{n-2\bar{\theta}^{n-2}}e^{-t/\bar{\theta}}t^{2-1}dt,$$
where $\bar{\theta}$ is the mean time between visits, estimated excluding the last two visits, and $n$ is again the number of visits minus 1. This is the (estimated) probability that the sum of the last two intervals would be lower than $t_0$ for that patient. If it is lower than 0.05, you can reject the null hypothesis and flag the patient.
Now, there is the question of the cut-off half-life parameter $\theta_0$. As an expert in your field, you are the only one to have an idea where the line is between FF and non FF patients, and this is the information you have to provide to the statistician. A way to get to $\theta_0$ would be to estimate the mean time between visit for patients that are certified non FF, ideally through confirmation by other experts.
These approaches are very approximate, but already challenging for a non statistician. My opinion is that this question deserves a better answer, but this requires some time. I would recommend you to setup a collaboration with an expert of time series analysis to use more state-of-the-art tools.
