My apologies for the belated answer, I had a very busy Red Dead Redemption work week.
This is a really interesting and complex question, but thankfully, it is one which is possible to address to some extent despite the sparse sources available. I assume you are familiar with the elementary features of Zoroastrianism - if not, I have a number of writeups on it here: https://old.reddit.com/r/AskHistorians/wiki/profiles/lcnielsen and feel free to ask followup questions on any of them.
According to most standard understandings of the origins of Zoroastrianism, it emerged in a pastoral society as a reaction against the destructiveness of bronze age warfare and its celebration in Vedic tradition. Deeply rooted parts of the liturgy, like the creed of Yasna 12, include statements such as "I will never again raid a Mazda-worshipper settlement", and closes:
I pledge myself to the Mazdayasnian religion, which causes the attack to be put off and weapons put down; [which upholds khvaetvadatha], Asha-endowed; which of all religions that exist or shall be, is the greatest, the best, and the most beautiful: Princely, Zoroastrian. I ascribe all good to Ahura Mazda. This is the creed of the Mazdayasnian religion.
The essential notion is that a purely martial society is evil (exemplified by the Vedic deity Indra, whose claim to lordship in a famous Vedic verse, Rv 4.42, begins "Men who drive swiftly, drawn by good horses, call on me when surrounded in battle...") - martial virtue must be exercised with Good Purpose [thought]. For example, khshathra, which can roughly be translated as "[the power of] rulership", personified as an amesha, receives the epithet "vairya", typically translated "desireable", in post-gathic liturgy. This then raises the question of what a good ruler is - clearly, merely using violence effectively would not suffice. Coincidentally, I wrote something on that this week, that will give some insight into what the tradition around Zoroaster's patron Vishastpa eventually became. It's unclear exactly when the core of the yasht dates from - it reminds me a bit of the idealized relationship between Aristotle and Alexander, which could be a possible influence on it.
Another interesting myth of good kingship is that of Yima or Jamsheed, most notably treated in chapter 2 of the Videvdad (it's short, I suggest you read it yourself). Yima is presented as a good shepherd, one Ahura Mazda asked to be a prophet before Zarathustra. Yima replies that he was not born to be a prophet, and so Ahura Mazda instead tasks him with ruling the world and making it flourish. Yima's rejection of Ahura Mazda's proposal is a bit odd; it may represent notions of caste division or perhaps the importance of individual choice in doing good. Yima rules over the world as a good shepherd, with perfect benevolence. Probably, this contains remnants of a very ancient Indo-European creation myth - Yima, usually taken to mean "twin", would then be cognate with Ymir of Norse religion, the primordial giant who is sacrificed or slain to create the world.
I think that will have to do as a far as mythology goes, especially given with how hard (read: impossible) it is to evaluate the antiquity of the Videvdad's material in most cases. We turn now to the Achaemenids. A few notes here:
I take Darius, not Cyrus, to be the founder of the Persian/Zoroastrian Achaemenid Empire as we think of it - partly because Cyrus is very obscure, and partly because what we know of him contrasts with Darius.
I take Darius to have largely founded the royal ideology he espouses (other arguments can be made, if one for example proposes the existence of a Median royal ideology before him)-
Many authors, including those whose judgment I trust (like Amelie Kuhrt) place little weight on the accusations of blasphemy found in Achaemenid inscriptions, assuming them to be propagandic rebranding of political discontent or minor rebellions. I believe they actually do reflect displeasure with e.g. Elamite religious practice.
I agree with more recent authors (again, like Kuhrt) that Darius' bizarre story of Gaumata the Magian is a fictitious one - it's a cover story for the fact that he rose to power by conspiring with other Persian noblemen to assassinate Cyrus' son Bardiya, and possible Cambyses as well, violently repressing the rebellions that resulted and making sure his absurd story was spread as the official narrative.
By far the most important text from the Achaemenid Empire is the Behistun Inscription which is tedious but worth reading if you have the time. I like Amelie Kuhrt's description of how Darius presents himself, as "the ruler of a large number of obedient subjects, each of which he governs with perfect justice". Darius hardly mentions a significant action without underscoring that he did it by the grace of Aúramazda or that Aúramazda bore [him] aid. The bizarre sequence of events leading up to his succession is presented in stark religious terms. Darius did everything to promote asha. His opponents did everything to promote druj. He was made king not by any human acknowledgement, but by Aúramazda himself (this is especially underscored in other inscriptions beginning with the stock phrase "Great is the god Aúramazda who created the earth ... and who made Darius king.") His mention at the end of Elamites who did not worship Aúramazda makes the divine nature of the Great King's authority clear - they did not worship Aúramazda, Darius worshipped Aúramazda, so he "bent them to [his] will" (temple tablets suggest a more diplomatic approach, where he sponsored temples to Elamite deities in exchange).
SImilar notions are repeated throughout the XPh inscription by his son Xerxes, who underscores that there was a land among one of twenty or so he mentions where daiva were worshipped, i.e. where the cardinal sin of Zoroastrianism was committed. In the crystal clear prose typical of Achaemenid inscriptions, he omits any mention of which land this was.
This serves as a basis from which we can cautiously begin to understand how the Achaemenid monarchs wished to be portrayed. Their actions were those of absolute justice, since their power derived from Ahura Mazda. By the same token, their power was unlimited, and exercised with perfect wisdom. What is repeatedly underscored is less the size of monuments or armies or piles of tribute, and more the righteousness of the Great King's rule, underscored even by regnal names like Artaxerxes (Arta-Khsaca, Righteous Rulership). This is both in line with a long tradition of Near Eastern kingship (e.g., Hammurabi receiving the scepter of justice from Shamash) and consistent with the fundaments of Zoroastrianism - the exception would probably be the extremely cruel execution methods Darius details, such as
I cut off his nose, his ears, and his tongue, and I put out one eye, and he was kept in fetters at my palace entrance, and all the people beheld him. Then did I crucify him in Ecbatana; and the men who were his foremost followers, those at Ecbatana within the fortress, I flayed and hung out their hides, stuffed with straw.
which probably violated more than a few purity rules, but these are not surprising considering the importance of the legacy of the methods of Ashurbanipal and other Assyrian rulers. Similar considerations would explain why Achaemenid Great Kings seem to have been interred, rather than exposed, upon their death.
Our ability to situate this royal image within Achaemenid society is unfortunately limited, which also means that it's difficult to say what the Great King's place from the point of view of the clergy was. In fact, such a point of view only becomes available to us with the inscriptions of the High Priest Kerdir (Kartir) of the Sasanian Empire in the late 3rd century. In addition to that, we have various texts such as the Testament of Ardashir (the founder of the Sasanian Empire, who claimed to be the grandson of the eponymous Sasan, a cleric) which unfortunately have extremely complicated textual histories but are nevertheless useful. A very pertinent adage comes from the Testament:
Know that kingship and religion are twin brothers, no one of which can be maintained without the other ... for religion is the foundation and kingship the pillar, and possession of the foundation has more claim than possession of the pillar.