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Unexpected results were attained in a simulation that point to a possible problem with the dataset. While writing about this in the discussion section, I'd like to elaborate on my explanation for this by presenting a simple calculation that reinforces my interpretation. Is it allowed to do this in a discussion section or should I lay the foundation for this in the method section and then include it in the results?

Generally yes. The purpose of a discussion section is to interpret your results, both in light of what was previously known (hopefully described in the introduction), and with new supporting arguments or hypotheses. A calculation (possibly short or rough) is a good way of providing a supporting argument, and it's quite common to see one in the discussion section of a scientific paper. However, theses are often subject to local rules, some of which may be silly, so make sure to check your institution's style or thesis guide, or check with your advisor whether they have any preferences.

I think it is fine and preferable. This really more "analysis" than "method" at least in the context of the original study design. Also for a reader it makes more sense in that order.

In general, there is some looseness about the exact configuration of discussion and results. As long as you are organized and show an understandable narrative, I highly doubt you will get someone telling you "that doesn't go in that section". I can look at papers in my field and see differences in how discussion was organized in any issue of the major journals and it is fine, no heads turned.

See also pages 8-15 of the attached: https://ocw.mit.edu/courses/media-arts-and-sciences/mas-111-introduction-to-doing-research-in-media-arts-and-sciences-spring-2011/readings/MITMAS_111S11_read_ses5.pdf