Well, after doing a bit of research, this is something we know surprisingly little about. We rarely know very much about the internal administration of the Achaemenid Empire, especially outside of the role of Greeks, but payment and provision is actually something fairly well documented. The vast majority of Achaemenid primary sources, and Near Eastern sources in general, are receipts, contracts, and other transactional documents. Notably, the Persepolis Fortification Archive regularly refers to the provisions given to various members of the royal family, nobility, and their subordinates. Unfortunately for your question, there are disappointingly, few references to any sort of military personnel.

Early on in Persian history, when Cyrus, Cambyses, and Darius were still building the empire all, or at least most payments of any kind would be handled in kind rather than currency. In the very early days of Cyrus's conquest, it just didn't exist. Gold and silver coinage was still a relatively new invention and had yet to spread beyond Lydia and Greece. After conquering Lydia, the Persians did take control of the mint at Sardis and continued minting Lydian-style Croeseids, but not in significant enough numbers to pay the whole army.

It's always hard to know how different payment in grain, or at least measured in grain, was from rations. Some examples seem clear, like PFA Tablet NN-1657, which documents a spear bearer for Xerxes (still a prince at the time) receiving one liter of flour for one day. That's an obvious bread ration. But other tablets document thousands of liters of grain or flour, which could rations for large groups, or significant payments, which had been common practice in Mesopotamia and Elam for centuries.

Later in the Achaemenid period, we know, that at least some soldiers were paid in coin, specifically the Greek mercenaries. In his Anabasis, Xenophon, a Greek mercenary himself, recorded that the 10,000 Greeks in service to Cyrus the Younger were paid approximately 1 Golden Daric (the Achaemenid currency instituted by Darius the Great). This may have been paid in 20 silver sigloi (the smaller Achaemenid denomination) over the course of the month, or in Darics at the end of the campaign. It's possible, that a similar payment system was used for other soldiers, given that a mint was also constructed in Babylon to produce additional Darics elsewhere in the empire. Just a few decades later, in his Hellenika, Xenophon reported that Greek mercenaries in Persia were being paid in Attic drachmai, which had greater buying power and acceptance at home than Persian sigloi. That said, darics were still used to pay Greeks and other mercenaries, as double-darics were produced in the reigns of Darius III and Alexander the Great. However, this would only have been useful for soldiers in or from the west. In the eastern part of the Persian empire, currency did not see widespread use until the Hellenistic period.

It's entirely possible that the majority of Achaemenid armies were not paid, and only received rations and board while on campaign. In this scenario, military service was considered part of a satrapy's tax burden. This was the case for the infantry during the much later Sassanid Persian Empire. If any portion of the Persian army was paid, it was certainly the so-called Immortals, described by Greek authors as a 10,000 man strong standing army, they were the closest thing to a professional Persian army.

The Immortals are also a good example to discuss, the last part of your question. As I've said, we do not have enough information to talk about pay figures in the Achaemenid military, but we do know about it's ethnic make up. The Immortals were entirely, or mostly Persians and Medes. Herodotus also makes reference to 1,000 "spear bearers" pulled from the young men of the nobility, and the spear bearer serving Xerxes in the tablet above was Parthian, possibly indicating that we can include the Parthians in the peoples permitted into the Immortals and Achaemenid nobility. Medes in particular account for a disproportionate number of generals outside the royal family. That said, the people of Parsa were explicitly superior to the subject peoples in the empire. They were exempt from monetary/precious metals taxes and made up the bulk of the occupying military forces.

After the Persians and the Medes, Greek sources identify the Saka as the next major component of the army. In Persian inscriptions, "Saka" refers specifically to the semi-nomadic peoples from the steppe, on the northeastern edge of the empire. In Greek sources, it's probable that the Bactrians, Arachosians, Sogdians, and other peoples in the eastern empire were identified as Saka as well due to their similar styles and cultures. The picture we're left with is a military that overwhelming privileged Iranians (or Ariya as they were called in Old Persian). This is probably because they fought and communicated more like the Persians than, and provided a significant cavalry component for the army.

In the navy, we see a very different picture, where Phoenicians made up the bulk of the crews and ships, with the Anatolian Greeks in second place, and Egyptians and Cypriots behind them. Occasionally riverboat crews from Mesopotamia and Central Asia were also drafted for service in the west. Persians and Medes were still always in command of the Mediterranean fleet. Presumably the Persians also had a naval presence in the Persian Gulf, but the only Persian naval activity outside the Mediterranean we have documented in our sources is Herodotus' account of Skylax, a Carian captain sent to scout the Indus River and chart a sea route between India and Egypt.

As for everyone else in the army, that whole famous list of peoples in different ethnic dress with unique weapons listed in Herodotus probably represents a nominal presence from all of the subject peoples in a major invasion force. It could be that they were there to demonstrate the scale and power of the empire, or that they were functionally hostages to ensure good behavior in their homelands while a large chunk of Persian forces were occupied elsewhere, possibly both. In the last century of the Persian Empire, Greek mercenaries made up an increasingly important component of the Persian infantry, but rarely to the same degree as the Iranians. The army of Cyrus the Younger is the obvious exception, given that it was a rebel army recruited in Anatolia. Of course, if the Persians did not pay their regular troops, then this would make the Greek mercenaries the highest paid component of the army, possibly second to the Immortals. Unfortunately, we just don't know enough to say anything with certainty, but just enough to provide plenty of speculation and plausible options.