I have read that it is common (though not a consensus) today for historians to date Zoroaster to the century or so before Cyrus the Great, in the 7th or even 6th centuries BC (around the time of the Median Empire), which would mean that Zoroastrianism, or his take on Iranian religion, spread very quickly. However, if Zoroaster was, or is defined to be, the author of the earliest Gathas, his language was extremely archaic within Iranian. What are the general arguments about when he lived, especially those arguing he lives as late as the 6th c.? And is ‘he’ potentially a prophet with a specific name, to be distinguished from whatever person/group authored the oldest parts of the Avesta? Or is it assumed that the Avesta was authored much later in language based on some archaic form preserved in the pre-Zoroastrian yasna rituals (with a clear Indo-Iranian origin), or that he happened to be from some lost tribe with a very conservative language?
As you alluded to in your post, modern linguistic evidence does not support a 6th Century date for Zoroaster, and few mainstream academics accept that date today. It has even fallen out of vogue in many Zoroastrian communities, who have embraced the social prestige of practicing an even more ancient religion.
The Avestan hymns, prayers, and scriptures - and the Avestan language by extension - was on of the earliest topics studied in comparative linguistics. Even the earliest 18th century linguists noticed the close relationship between Sanskrit and Avestan, which formed the initial basis for the idea of an "Aryan" language group, which we now call Indo-European, named for how both languages used "Aryan" to describe the in-group of the authors.
Linguistic study has come a long way since that initial connection. There are now established, predictable patterns of change that can be used to date how much time passed between stages of a language, diversion from a parent language, how written vs oral traditions effect a language, and even how to date a languages influence on another language. Much of this is part of the field of quantitative linguistics. Using these patterns, it becomes clear that the Iranian languages split from the Indo-Aryan language that produced Sanskrit no later than 1500 BCE, and the Eastern and Western Iranian languages split apart not long after that. From there, comparisons to later Eastern Iranian languages like Sogdian or Bactrian, and it's influence on Western Iranian languages like Old and Middle Persian help date Avestan.
Avestan itself was easily identified as two distinct phases or languages depending on how you want to think of it. It can be loosely compared to Middle vs Modern English. These two phases are called Old Avestan and Younger Avestan. The latter makes up most of the Avesta and linguists have identified different phases within that corpus. Comparison with other languages and predictable changes put Younger Avestan as a spoken language between 900-400 BCE. Old Avestan is thus the much smaller corpus that predates 900. In order of age, it includes the Gathas, Yasna Haptanghaiti, and the "Five Sacred Prayers." The prayers are just a few lines each, and contain the early elements of Younger Avestan grammar. The Yasna Haptanghaiti is also fairly short and appears to be a slightly younger variant of Old Avestan than the Gathas, possibly incorporating an even older verse in the middle.
The Gathas are thus the oldest section of the entire Avesta, dated to around 1500-1000 BCE. Given the apparently rapid shift from the Old Avestan of the Gathas to the Younger elements of the Yasna Haptanghaiti, more and more authors are leaning into the latter half of that range. Not coincidentally, they are also the section of the Avesta traditionally attributed to Zoroaster himself. I'm going to answer the second question first, since we're on the topic here. The Gathas are very consistent in terms of style and content, so much so that they are generally believed to be the product of one author, or at least a very organized small group. They also portray the speaker as calling himself Zoroaster in multiple instances. That's about as close as you can get to identifying a specific author for a Bronze Age oral tradition.
That is the information and the process used by most modern scholars to assert a c.1000 BCE date for Zoroaster.