Just on the pitta/pīta issue, these are, from the point of view of historical phonology, alternants of the same word meaning "yellow" (cf. law of morae; Mayrhofer). Bile, pitta, in Indian medicine, was "[yellow, pīta] bile." The word pīta occurs very frequently in ayurvedic sentences in close collocation with the word pitta (e.g., see this
listing of collocations). So I think this semantic connection was alive in the minds of ayurvedic authors. This deserves proper study.
India did not evolve the concept of "black bile" (Gk. melancholia) that evolved in Hippocratic medicine.