Short answers, pulled from the story.
The Middle Way refers to a path that steers clear of two extremes: sensual indulgence and self-mortification. It means entering neither of these extremes as stated in the Dhammacakkappavattana Sutta.
The Buddha delivered his first teaching, the Dhammacakkappavattana Sutta, after achieving awakening. This event marked the beginning of the path that avoids the two extremes of sensual pleasure and severe asceticism.
Nagarjuna set forth this philosophy based on Buddha's Perfection of Wisdom Sutras. Aryadeva was Nagarjuna's student whose work Four Hundred Stanzas on Middle Way explains meaning of Nagarjuna's work including refutations of non-Buddhist systems.
In the Theravada Buddhist tradition usage of the term Middle Way is discussed in fifth-century CE Pali commentaries. The influential Theravada doctrinal compendium called the Visuddhimagga interprets metaphysical import differently among modern Theravada Buddhists.
Gelug school upholds Rangtong Empty of Self philosophy arguing emptiness absolute negation meaning everything including Buddhahood emptiness itself said empty. Shentong Empty of other philosophy mainly propounded non-Gelug Tibetan schools holds ultimate reality only empty impermanent conditioned not empty own true nature.