1) Doma is a mix of Chinese and Japanese cultures. I don't know how the inference to Nara Japan took root, but to me it's most definitely wrong, because such an analogy doesn't account for the look and feel of Yanxia as a whole. The naming conventions, customs and costumes of the Doman people are certainly Japanese in inspiration, but the landscape and architecture is most certainly Chinese. Yes, I know Nara Japan was consciously modelled after Tang China, but geomancy and the Qitian Dasheng, aka Sun Wukong, in the Swallow's Compass are decidedly Chinese, not Japanese. Even the name, Yanxia, is Chinese (it means the Swallow's Gorges). Hinganshi is much easier to place, as its analogy with the early Edo Shogunate is very easy to spot.

2) Courtesans existed in all cultures, and certainly in Chinese and Japanese history. But, as far as I know, it was mainly in Japan that they are seen as a distinct professional class deserving of a certain level of respect. Geisha clearly operate in Kugane's pleasure and theatre district. As for Doma, well, that was the unfortunate fate of Yotsuyu, who was made to do more than just "entertain" guests.

3) Thaumaturgy is distinctively Ul'dahn. But other cultures will likely manipulate aether in similar ways.

4) As suggested, it's a good bet you can refer to raen naming conventions for hyuran Hinganshi surnames.