Hi @AnnaC,
You could use both. The regular change in diet certainly complicates matters, so on the one hand the pairwise comparisons would be a better approach for isolating the effects of specific treatments.
On the other hand, LME may be appropriate, depending on your treatments and hypothesis. Are the changing diets a gradual introduction of some ingredient and you expect a gradual increase in, say, alpha diversity, or is this a cross-over study and you expect the changing diets to reverse the effect? If the former (or in any case where you expect or see a linear change), LME would be useful, and you can "eye-ball" how well LME may fit your data by using the volatility action to look at the shape of the data. If the latter (cross-over), then your data probably aren't linear, LME would make a poor fit, and paired comparisons would be a better choice.
Sure! LME is a really useful method for longitudinal experiments but it is not in itself an inherently longitudinal method. This is why the parameter name state is used throughout q2-longitudinal instead of calling it "time"... because many of these methods can be applied to both longitudinal as well as other experimental designs that involve dependent samples.
I suppose here's one caveat: assigning numbers arbitrarily may make too many assumptions about the relationship between these sites... I'd feel more comfortable using a quantitative measure (e.g., distance from surface), but I suppose you could use volatility to see what the data look like after assigning those labels. Given that you only have 3 dependent samples, and especially because it sounds like there could be methodological differences in how they were collected (surface swab vs. skin biopsy?) pairwise comparisons may be more appropriate in this case.
Good luck!