Is it meaningful to talk about single-event frequentist probabilities? When learning about confidence intervals, we are told that we must not talk about "the probability that the true value lies within the interval" because (frequentist) probability is the limit of frequencies in a mostly-fictional population of experiments, and has nothing to say about whether a fixed value lies in a fixed interval.  (Except, perhaps, to unhelpfully say that it is either 0 or 1.)
Nitpicky note: I am aware that if we consider the confidence interval to be a random variable then a useful probability statement can be made about it - but whenever I calculate one from real-world data, I seem to get a fixed interval and not a random variable. :D
But...
1 in every 100 million people have the brain-exploding flu.  A man is tested for BEF - with a test that is 99% accurate in both directions - and the test results come up positive.  The man asks his doctor if he really has brain-exploding flu.
I feel like the doctor should be able to say "almost certainly not".  Just eyeballing it, the probability is about 1 in a million here... or is it?  Doesn't the same objection apply?  This is also a single event, and whether the man has BEF is an unknown but fixed value.
 A: In your example, the test does not tell us the risk of disease. It makes a decision. It is either a true positive, true negative, false positive, or false negative. The doctor, in trying to explain the test, would tell the patient that the results were a false positive. The patient, not having died from the disease, is a living testament to the imperfection of that diagnosis. Diagnostic tests are unconcerned with ascribing a single event probability risk.
In medicine, we differentiate between diagnostic tests and risk predictions. Risk predictions use a patient's risk profile to construct an time-interval based risk prediction given you are at-risk for disease at that time. For instance, the Gail model predicts 5 year breast cancer risk in a woman who is free of breast cancer and has both of her breasts (no mastectomy; no breast tissue=no cancer risk). Age, menopausal status, family history, and other risk factors are reported. The model gives a percentage breast cancer risk. This percentage is used to counsel the patient and advise preventative options. 
A Bayesian interpretation of 5 year breast cancer risk is readily accepted. A frequentist relies on the concept of potential outcomes. These are like counterfactual inference. We presume women who are otherwise similar in their risk profile pose a type of superpopulation whose frequency of 5 year breast cancer incidence is concordant with the risk prediction.
A: This would fall under the philosophy of probability imo. I would suggest you start with the SEP entry on interpretations of probability and then possibly continue with following the references therein. 
A: Yes. Also "frequentist" and "fictional" come together naturally only in social sciences. (Physicist Lev Landau joked that there are three types of sciences: natural, unnatural and antinatural).
You don't need to use a word Bayes even once in this 19 pp paper to come up with a significance number for a single event: Observation of a first $\nu_\tau$ candidate in the OPERA experiment in the CNGS beam, N. Agafonova et al. (OPERA Collaboration) arXiv:1006.1623 [hep-ex]
It's a rather technical article in high energy physics, so in this case it's Ok to jump to Conclusions section (pun intended):

The observation of one possible tau candidate in the decay channel
  $h^-(\pi^0)\nu_\tau$ has a significance of 2.36 $\sigma$ of not being
  a background fluctuation. If one considers all decay modes included in
  the search the significance of the observation becomes 2.01 $\sigma$.
  The expected number of $\nu_\tau$ events detected in the analysed sample is
  0.54 $\pm$ 0.13(syst.)

