One of the things I've been interested in while looking at the Achaemenid Empire has been the cultural/ethnic groups that have lived inside of it. Such an empire would've had many different people groups; like the Babylonians, the Egyptians, the Persians, the Jews, the Bactrians and the Ionians. But there are a few questions I'd like to be answered about this subject which have always bugged me:
Who were the main Ethnic Groups of the Achaemenid Empire asides from the ones mentioned?
Besides the one you mentioned, there were:
Medes, quite closely related to Persians.
Armenians, who had apparently overtaken the Urartians at this point.
Assyrians, closely related to Babylonians
Various Central Asian Iranian groups.
Various Indic groups.
Anatolians (e.g. Lydians)
How did this affect the organization of the Achaemenid Empire?
I'm not exactly sure what you mean. Our understanding of Achaemenid administration remains somewhat limited, but we can be pretty sure that the Satrapial system worked as a quasi-feudal institution wherein the Achaemenid house managed to monopolize political legitimacy, such that marrying into the Great King's house and gaining his favour was the most viable path to political power. It's hard to say if the particular provincial divisions of the empire were due to necessity, ideology, or its multiethnic nature. The Behistun Inscription and other sources certainly suggest that Dareios' divisions were based on ethnic lines, but also that he portrayed himself as a king of many lands in addition to king of Persia, rather than as king of a great empire. However these types of phrasings also have antecedents among Mesopotamian rulers, making it hard to make heads or tails of them.
Were there any attempts at a sort of ethnic/nationalist revolt in the early years of the Empire; ie the Ionian Revolt?
Egypt's elites were more or less continuously attempting to assert themselves. The Persians seem to have maintained decent control up until the rebellion of Cyrus the Younger, when they effectively lost control of Egypt for decades.
Why didn't these ethnic/cultural groups attempt to make their own nation-states after the death of Alexander the Great's conquest of the Achaemenid Empire during the Wars of the Diadochi?
The concept of a "nation-state" is an invention of the past few centuries. Yes, it seems natural to us because we are constantly drowned in ideological messages and justifications on the basis of the nation-state being the natural order of things, and ridiculous back-projections of national identities over thousands of years, but for almost all history, ethnicity and nationality were far more fluid things than we think today. Depending on who you were and when, your primary allegiance and might be to your city, to your clan, to your wealthy and powerful in-laws, etc.
It is hard to speculate on why there were not more succesful independent minor states, but the one that seems most natural to me would be in part about economies of scale - for many communities (e.g. ones which lacked the social stratification to have a wealthy military elite), it was simply far more viable to pay tribute to a distant king in exchange for protection than to organize their own.