I have a previous answer written on the subject of perceptions of Alexander in post-Hellenistic Persia: https://www.reddit.com/r/AskHistorians/comments/ahuixh/why_is_alexander_the_great_remembered_in_history/ where /u/Iphikrates adds additional insights with more nuanced views on Alexander from the Hellenic side of things.

To address your questions to some extent:

• Claims of universal dominion are pretty common in ancient imperial societies in e.g. the Near East and China. However, they are more often than not paired with more realistic assessment of the nature of things. The Achaemenids post-Xerxes seem to have been pretty content to not exercise direct rule over either Greece or the (similarly distant) city-states of the Gangetic Plains in India, lacking an effective mandate for exercising hegemony over either. EDIT: Another example from a slightly later era would be Roman-Sasanian relations. The Sasanians claimed Rome as a vassal state on the basis of Shapur's humiliation of three emperors, whereas Rome made its usual assertions of imperial legitimacy. At the same time, the only monarch the Romans recognized by the Greek title Basileus, besides the Roman Emperor, was the Sasanian King of Kings; conversely, in Persian, the Roman Emperor was acknowledged by the special title Caesar, and the Chinese Emperor by the special title Bagpur, literally "son of God", a calque of Chinese Tianzi, "Son of Heaven".

• It's a complicated and heavily ideological question that will vary greatly from society to society. Perhaps the most important question is whether the would-be conqueror was thought to possess a mandate to conquer the world. For instance, classical sources on Cyrus the Younger's rebellion against Artaxerxes II, especially Xenophon, portray Cyrus as justified in his endeavour on the basis of his noble character and personal virtue being more suitable for a Great King. Or consider all the figures vying for hegemony in the Three Kingdoms era in China, and how differently e.g. Cao Cao, Dong Zhuo and Liu Bei are portrayed, despite trying to accomplish by and large the same thing (hegemony over Tianxia, "all under heaven") by similar means.

• In terms of relative political power, Achaemenid dominion persisted in a much more monopolar world than would e.g. Sasanian, Roman, etc dominion, and the Achaemenid Empire was really completely unmatchable in terms of political, military and economic power, at least until the various minor powers around its borders began to coalesce into coherent political entities, culminating in Alexander's invasion. Whether that is worth thinking of as "world dominion" is an unanswerable question.