I cannot speak too well on the Greek situation, however on other points:
The Babylonian kingdom however was reaching an increasingly polarized situation, yet also a period of a potential Akkadian renewalism.
Nabu-Na'id or Nabonidus (556-539 BCE) was a powerful king in the image of his older predecessor some 40 years prior, Nebuchadnezzar II (605-562)(Naboo-kuduri-usur II), he made military gains against the Neo-Qedarites to his south, the Nabtu and likely ruled the Gerrhea coastline of Arabia. Likewise, his rule coincided with a renaissance of Akkadian linguistic exercise and a renaissance of classical Akkadian cultural motifs. Most formally, the Babylonian renewed interest in the restoration of varied temples, the excavation of many reliefs, statues and so forth. Many of these innovations were already at hand in the Assyrian empire, which fell around 60 years before the ascension of Nabonidus, however its return to the forefront was a sign of possible recuperation for the Akkadian universal empire motif.
Nabonidus himself appears to have been a reformer of some kind, both religiously and secular. His devotion to Sin (the moon god) is critical as was the revival of titles he came to take, that had been absent in the region since around 1890 BCE. Namely, Nabonidus, began to refer to himself as the God of the Land and God among the Gods and so forth. This led to him becoming an outcast in the city of Babylon and thus he spent more time in his Arab holdings, Sumer or in his hometown, Harran in the north, the cult centre of Sin. Nabonidus also was noted for his many reliefs wherein he claims his power and authority over the lesser Gods, his devotion to Sin and his majesty, mimicking the Assyrian kings preceding him and the more ancient Kassite period of Babylonia or Nebuchadnezzar I (1134-1108 BCE).
Without Cyrus, the Babylonian kingdom is still in a weak position, even with Nabonidus.
1. It is divided at critical moments. The city of Babylon is wavering in its support to Nabonidus and likewise to the wavering dynastic royalty, who increasingly were moving their capitol region northward to Harran and creating a southern winter capitol in Tima (the lands of the Nabtu in Arabia).
2. The kingdom of Babylon in this era, was no Assyria. They ruled only the southern districts of Assyria, Suhum and Shupria. Assyria proper was more or less ravaged in the prior six or seven decades. Babylonian Shupria was more fortunate, Harran remained a major city and succeeded the city of Ninevah as the primary city in the Akkadian northern world. Despite this, Shupria was generally decimated by the prior decades of Scythian and Median incursion and lacked the same manpower for wars as in the Assyrian period prior. Babylonian armies also lacked the Assyrian systems to gather such large armies. Suhum was likely in decline also, though it was not decimated as heavily.
3. The current Babylonian kingdom was ultimately a non-Babylonian dynasty. They were certainly of the Aramaic class of people inhabiting the region called the Chaldaens and were closely associated to the more eastern Gambulu confederates. These Chaldaens were likely supported from an early period by Elam and the prior iterations of the Babylonian kingdom that rebelled against Assyria, were little more than an Elamite vassal or client state. It was analogous to a series of Elamite supported rebellions that due to the Elamite proximity to the area (closer than Assyria) and their affiliation with the Gambulu and Chaldaens, allowed the Elamites to establish consistent rebellions in the area. The drawback to this situation, was that the Elamites were both the military and diplomatic collateral and overlord in these interactions, fighting Assyria on behalf of these Faux-Babylonian kingdoms. It left the Babylonians a power that was more hollow, if you will, than the Assyrian predecessors.
The main question, is if Nabonidus could muster the Babylonian kingdom into a more powerful realm that would resemble the Assyrian predecessors. It is certainly possible, but difficult to achieve. Admittedly the powers around it, are not extremely powerful either. The Medes made much of their gains against Assyria solely due to the massive coalition arrayed against Assyria and likewise the Assyrian instability since the death of Assurbanipal. Even with all of these balances in favor of the Medes, they barely were able to defeat Assyria after around 15 years of consistent battle across Assyria and then Shupria, as Assyria seems to have harried the coalition numerous times alone and weakened.
