Short answers, pulled from the story.
Neoplatonism began with Ammonius Saccas and his student Plotinus in the 3rd century AD. Plotinus, who lived from around AD 205 to 271, is widely considered the father of Neoplatonism; the tradition stretched to the 6th century through his successors.
"The One" is the first principle of reality in Neoplatonism: an utterly simple, transcendent source from which all of reality emanates. It is beyond being and non-being and cannot properly be said to exist in any ordinary sense; the most adequate names for it, according to Plotinus, are "the One" or "the Good".
Neoplatonic texts became available in Arabic and Persian translations after Muslims conquered Byzantine centers in Egypt and Syria, where Greek manuscripts were held. Thinkers including Avicenna, al-Farabi, al-Kindi, and Ibn Arabi adapted Neoplatonism to Islamic monotheism, identifying the First Principle with God.
Hypatia, who served as head of the Platonist school in Alexandria around 360-415, was murdered by a mob of Coptic Parabalani monks. She had been advising the prefect of Egypt, Orestes, during his feud with Cyril, Alexandria's archbishop, and the extent of Cyril's personal involvement in her death remains debated by scholars.
Marsilio Ficino translated all 36 of Plato's dialogues into Latin between 1462 and 1469, after Cosimo de' Medici provided him with the Greek texts. Between 1484 and 1492, he also translated the Enneads of Plotinus, making Neoplatonic works available in Latin for the first time in the West.
Neoplatonists believed in the pre-existence and immortality of the soul, as well as reincarnation. Plotinus held that a soul could be reincarnated into another human or even an animal, while Porphyry maintained that human souls returned only into other humans. A soul that achieved union with the One would not descend again.