What kind of rice should I use to make Persian/Indian/Middle Eastern rice-based desserts?
(Disclaimer, since I don't read/speak Farsi, I only looked at English references) 

Authenticity is in the eye of the Beholder. 

Iranian and Indians cooks probably use whatever rice is available in their local store; and some will use the same rice variety that their mother and grandmother used without knowing the reason.

You can look at this which talks a little bit about rice in Iran, so does the wikipedia page about Iranian cuisine.

For example, the Domsiah is related to Basmati rice.

That being said, 

Most recipes for shole zard, as you say, just say white rice, or either Jasmine or Basmati rice; I've also seen recipes with short grain rice; so pretty much use what you have on hand.

For firni, most recipes call for Basmati rice which looks to be most common in India.