Short answers, pulled from the story.
Roma quadrata was an area or structure within the original pomerium of ancient Rome, most likely the Palatine Hill including its Palatium and Cermalus peaks and their slopes. The Latin phrase means "square Rome", with the Greek equivalent Tetrágōnos Rhṓmē carrying the same meaning. It apparently dated to the earliest stage of Rome's formation.
Roma quadrata was probably located on the Palatine Hill, encompassing both its Palatium and Cermalus peaks together with their slopes. It fell within the original pomerium, the sacred boundary line of ancient Rome.
Four ancient authors left key passages on Roma quadrata: Dionysius of Halicarnassus in his Roman Antiquities (Book II, chapter 65), Plutarch in his life of Romulus (Parallel Lives, Book 1, chapter 9), Tacitus in his Annals (chapter 12, paragraph 24), and Varro as cited by Solinus in the Polyhistor (Book 1, chapter 17).
By the late Roman Republic in the second century BC, the original meaning of Roma quadrata had already become obscure to both Latin and Greek historians. The term apparently dated to the very earliest stage of Rome's formation, meaning its meaning may have begun fading centuries before the late Republic.
The Murus Romuli refers to the walls of Roma quadrata. It is a related structure that scholars have traced alongside the textual evidence for Roma quadrata itself.
Andrea Carandini wrote about Roma quadrata in his 2006 Einaudi book on Romulus and Remus, placing it within a chronological framework of roughly 775-750 BC to 700-675 BC. He also dedicated a chapter titled "The Blessing of the Palatine and the Founding of Roma Quadrata" in his 2011 Princeton University Press volume Rome: Day One, pages 50-62.