Short answers, pulled from the story.
A vassal state is any state that has a mutual obligation to a superior state or empire. This status mirrors the relationship of a vassal in the feudal system found in medieval Europe.
The reign of Thutmose III spanned from 1479 BC to 1425 BC. Vassal states in the Levant became fully integrated into Egypt's economy through harbor construction which allowed for greater communication and tax collection between Egypt and its vassals.
Relations peaked under Šuppiluliuma I and Muršili II in the 14th century BC when terms were imposed unilaterally by the Hittite king and accepted by the vassal ruler. A new treaty was drawn up whenever a new king came into power and often involved marriage between a vassal ruler and a Hatti princess.
Tiglath-Pileser I ruled from 1114 BC to 1076 BC during the Middle Assyrian Period and initiated the collection of flora and fauna from vassal states as an expression of imperial hold. Artistic representations show subjects bowing or crouching before the king with gifts ranging from horses and monkeys to wineskins.
Herodotus records negotiations between King Amyntas I of Macedonia and Persians after subjugation by 513 BC. Under Darius I Macedonia organized into a regular tax district per Hdt. 6.44 and DNA inscription at Naqsh-I-Rustam attests to their control.
The Qing dynasty viewed Joseon Korea as an autonomous vassal state that remained independent in internal and external affairs without becoming a colony or dependency. China abandoned noninterference policy toward Korea adopting radical interventionism in late 19th century while William W. Rockhill stated calling Korea a vassal state was misleading.