A. However, one question did pop into my head - are these standard
tests based on the frequentist philosophy or the bayesian philosophy?
They are grounded in frequentist reasoning. Nonetheless, while Bayesians do not focus that much on hypothesis tests, as already noted by kjetil b halvorsen, but often there are Bayesian alternatives to the tests, for example BEST as an alternative to $t$-test. This is covered to some extent in Doing Bayesian Data Analysis by John K. Kruschke.
B. Also, as of now I view frequentist and bayesian views as mutually
exclusive with their own sets of non-overlapping theories. Is this
correct? Or is there an overlap in certain problems where one is
clearly better than the other (one uses more known values than the
other OR has lesser assumptions OR does more fine grained
calculations, for example). (in other words: bayesian theory looks
daunting - why would I learn it if frequentist works fine?)
They differ. This does not directly mean that one is better then another. First of all, they differ in their interpretation of probability and this has consequences in other areas. TLDR; Bayesian interpretation of probability is less restrictive and more general, what enables Bayesians to think in probabilistic terms of things that are usually not interpreted in such fashion. There are arguments on which interpretation of probability is more "intuitive", but at least in some cases Bayesian approach gives more intuitive interpretations, e.g. in case of confidence intervals vs credible intervals. You can check other questions tagged as bayesian to see extended discussion on this subject. As about assumptions, both approaches can make multiple assumptions. One of the differences in here is that in Bayesian case you are forced to state your a priori assumptions about parameters directly in the model, while in frequentist case they may not be stated directly but follow from the used procedure.