The Achaemenid Empire was an incredibly vast and diverse political entity (and not a "country" in the modern sense, but rather a collection of regional political authorities under the hegemony of the Great King of Persia - the most powerful man in the world at the time), and it is very difficult to enumerate a set of "religions" practiced in it for several reasons. First, "religions" did not come in the neatly delineated boxes that we often imagine them today. I say imagine because I don't think it's true that religion is neatly delineated in the 20th century, though many religions certainly claim to be. But back then, it wasn't really even the case that religions claimed to be delineated, as a rule. Religious practices were not clearly distinct from ethnic belonging, and the practices and beliefs of other groups were compared and sometimes equated to indigenous practices. For instance, Herodotos notes that "the Persians worship Zeus, who they equate with Ouranos [the sky, grandfather of Zeus in some Greek myths]." As far as we can know, the Persians regarded this as sensible enougho as Sasanian inscriptions indeed has the Persian Ohrmazd calqued as "Zeus" in Greek.

With this in mind, what do we actually have? You mention Egypt and early Judaism; to the latter I would add Canaanite religion in general in the vicinity of Syria and Palestine. This was at the tail end of a transformative era when we see "the end of the beginning" of Judaism as we think of it starting to form under Achaemenid dominion, although the process would not "complete" in a sense for perhaps a thousand years.

In what is today Iraq, we have a range of Mesopotamian religious systems. I'm not sure I can do the matter justice here, but they generally center on the worship of a vigorous warrior deity, sometimes described as a storm god, (Marduk, Ashur) at the head of a pantheon of other recurring figures, such as a solar deity (Utu-Shamash) of law, justice and kingship. They have things in common and interaction with but also distinctions from Indo-European religions.

There must also still have been Anatolian religion practiced in Eastern Anatolia, namely Lydian religion. These were Indo-European religious systems influencing and influenced by Mesopotamian religion; as far as I know most of our knowledge is about forerunners such as the Hittite religion, and there one cannot easily construct an overview as gods seem to be largely local and our sources are overwhelmingly elaborate ritual texts (very interesting in their own right, of course, as religion in this time was more about practice than belief). In any case, Hellenization was already happening here.

In Armenia there must have already been Iranization going on as Armenians were considered "Iranian" in the post-Hellenic era, so they practiced some variant of Persian religion.

Which brings us to my point of main expertise - Iranian religion. The Achaemenid kings repeatedly describe themselves as worshippers of Mazda, which is the endonym for what we normally call "Zoroastrians" in English. How Zoroastrian exactly they were, and exactly what that means, is a matter of contention that I personally regard as a bit of a red herring. Rather it is probably better to understand Zoroastrianism as a religious system that while containing certain basic precepts such as the supremacy of the creator Ahura Mazda, the central value of Good/Righteousness/Truth as opposed to Evil/Bad/Deceit, was always a dynamic system, able to adapt to and absorb local traditions while maintaining its central characteristics (compare this to various schools of Buddhism today). The Persian kings were notoriously able in portraying themselves and their divinely sanctioned rule within the context of the religious systems of whatever people they conquered (e.g., Dareios I describes himself as a god to the Egyptians). It's sometimes argued that in the later era of the Empire, under the rule of Artaxerxes II, Zoroastrianism became more formalized and adopted as a state religion in the imperial core, though evidence for this is sparse and indirect. However, everything points toward many aspects and oral traditions of Zoroastrianism that would later come to be viewed as integral to it were ossifying in the Achaemenid era, having been more fluid earlier.

In the East of the Empire, Iranian groups in Central Asia probably practiced some form of Zoroastrianism or other. In what is today Pakistan and Northern India, we would have been seeing the proliferation of Buddhism and Buddhist syncretisms and early reactions toward it by what we might call proto-Hindus/Jains/etc. The interaction between these, Mesopotamian religion and Hellenic religion brought by Alexander and his successors would come to define the dynamic and multifaceted religious environment of Central Asia and Northern India for centuries, arguably even a milennium.

I don't think I forgot any major group; I hope this answers your question and feel free to ask any followups.