By the time of the Persian kings that most people have heard of (Darius, Xerxes, etc), Zoroastrianism was well-established as the official religion of the empire. We know this is likely to be the case for a number of reasons; most prominently, the official proclamation commissioned by Darius at Mount Behistun, in which he recounts his many successful conquests by the favor of the Zoroastrian God, Ahura Mazda.

Beyond that basic point, the short answer to many of your questions is, unfortunately, that we don't know with a great degree of certainty. Most collections of tablets and inscriptions from the era provide little insight into daily life. The most extensive writings on the subject from the period itself are those of Greek writers - such as the famous Herodotus - whose accounts, while insightful, should not always be taken as word for word "fact" in the way that we use the concept today.

The actual practices of the religion at that point are similarly ill-defined. There are not any systematic records of the practices or beliefs of the Zoroastrianism at that point, and consequently scholars are left to try to piece together the fragments they have as well as possible - with predictably variable interpretations.

This notwithstanding, Mary Boyce, one of the more prominent voices in the discussion, wrote History of Zoroastrianism in which she asserts that the religion of the Achaemenid period was in fact quite similar to the version that was more rigorously recorded in later centuries. This is a hotly debated thesis within the field, but it is a fixture in the historiography.

Archaeology is critical to the discovery of more information; the Persepolis Fortification Tablets were discovered in a dig back in the 1930s, and those discoveries shaped much of what we know. More information would likely have to come from similar discoveries. In terms of interpreting what information there already is, linguists are at the center. Currently the debate centers around interpreting the evolution of language in the region, attempting to divine more about daily life. Those are two fields one could look into further either to get a sense of the nuance of the debate, or to eventually contribute to it.