Short answers, pulled from the story.
Buddhist philosophy is the ancient Indian philosophical tradition that developed within Buddhism following the death of Gautama Buddha around the 5th century BCE. It encompasses rational inquiry, epistemology, metaphysics, ethics, logic, and the philosophy of mind, and spread across Central Asia, East Asia, and Southeast Asia as Buddhism expanded from its origins in northeastern India.
The Buddha defined the Middle Way in two related senses. In the Dharmacakrapravartana Sūtra, it refers to steering between extreme asceticism and sensual indulgence. A second sense, which became central to later Buddhist metaphysics, describes a metaphysical path between eternalism and annihilationism, and between existence and non-existence.
The doctrine of non-self holds that no part of a person is unchanging or permanent. The Buddha argued from the five aggregates of existence (form, feelings, perceptions, volitions, and consciousness) that since all five are impermanent, and since there is no more to a person than these aggregates, there is no fixed self. This is expounded formally in the Anātmalakṣaṇa Sūtra.
The Buddha left questions such as whether the universe is eternal, whether the body and self are unified, and whether a person ceases entirely after death deliberately unanswered, calling them the "unanswered questions" (avyākṛta). In the Aggi-Vacchagotta Sutta, he described such speculation as "a thicket of views." His stated reason was pragmatic: these questions contribute nothing to the practical goal of liberation from suffering.
Nāgārjuna, active around 150-250 CE, gave the Mahāyāna doctrine of emptiness its most rigorous philosophical form in his magnum opus, the Mūlamadhyamakakārikā (Root Verses on the Middle Way). Using reductio ad absurdum arguments, he attacked any theory assuming inherent essence or independent existence, and asserted a direct connection between dependent origination, emptiness, and non-self.
Yogācāra is a Buddhist philosophical school associated with the brothers Asanga and Vasubandhu, arising between the 2nd and 4th centuries CE. Its central concept is vijñapti-mātra, or "impressions only," which holds that we only ever have access to our own mental impressions and cannot validly infer the existence of external objects from them. The school also introduced the doctrine of the storehouse consciousness (ālayavijñāna), a subliminal mental stratum that carries karmic impressions.