This is quite complicated, so bear with me for a bit! First of all, it is important to note that all these biographies of the Prophet Muhammad date to the late eighth century at the earliest and that these letters to various leaders of the world around Arabia are not taken as serious evidence for his contact with the Roman/Persian/Axumite Empires. However, Muqawqis can be identified with a real historical figure - the Chalcedonian Patriarch of Alexandria between 630-42, Cyrus of Phasis. To set the scene a bit, Egypt was largely Christian, but it was divided between the Chalcedonian minority (named after the Council of Chalcedon in 451) and the miaphysite majority, who opposed the doctrine promoted at Chalcedon. The imperial government based in Constantinople was generally pro-Chalcedon and in 630 the emperor Heraclius imposed Cyrus on the Egyptian patriarchate. Cyrus was from the Caucasus, explaining his name in the Muslim sources - Muqawqis means 'the Caucasian' (or at least, that's what I've read, I don't know any Arabic). He essentially acted as both the secular and religious leader of Egypt, since he was basically the governor of Egypt at the same time, explaining perhaps his depiction in later Muslim sources as an important leader in Egypt. He certainly proved to be crucial in the events of 640-2, as it was under his rule that the Arabs invaded Egypt. To make a long story short, by late 641 the Roman military situation was hopeless and Cyrus helped to negotiate a deal with the Arabs: the Romans would hand-over Egypt to the Arabs peacefully by September 642.

For your other questions, it is unfortunately quite difficult to answer. We know very little about the seventh century and the rise of Islam, so it is hard to pin down exactly what happened. Cyrus was treated horribly by later Greek/Coptic sources, as he was not only seen as a heretic by the miaphysites, but also by the Chalcedonians, since he followed a branch of Christianity now termed monotheletism. In one ninth-century Greek source, the Chronicle of Nikephoros, it is indeed recorded that Cyrus sought to make a deal with the Arabs before the invasion of Egypt, even to the extent of offering an imperial bride to 'Amr, the future conqueror of Egypt. This matches (to an extent) the story you related here, so it is worth thinking about whether the two late sources, Nikephoros and the Muslim tradition, may reflect a shared awareness of an earlier attempt at appeasement by Cyrus. It is also worth noting that Muhammad's death in 632 is debatable, as all the early sources imply that he personally led the invasion of Palestine. If this interpretation is correct, it is certainly plausible that Cyrus had some sort of diplomatic relations with Muhammad when Arab forces entered Roman territories. At least one other Roman commander on the Mesopotamian frontier was willing to abide by a temporary peace treaty with the Arabs, so it is not impossible that Cyrus did the same. I'm not too familiar with the later Muslim historiography, but are the stories of Maria and Ibrahim recorded in the work of Ibn Ishaq, the first biographer of the Prophet? If they were, I would say that their historicity is more certain. If they were later, such as from the compilation of al-Ḥakam (who wrote the first Arabic history of the conquest of Egypt), I would be more wary, since later Muslim traditions were rooted in legal traditions of the ninth century rather than the historical reality of the seventh century, though obviously there must have been a kernel of historical truth even within these later accounts.

For further reading (and I know some of the stuff I talked about here might be a tad controversial), it is worth reading the sections on the conquest of Egypt in James Howard-Johnston's Witnesses to a World Crisis: Historians and Histories of the Middle East in the Seventh Century (2010) and the very brief description in Robert Hoyland's In God's Path: The Arab Conquests and the Creation of an Islamic Empire (2014). For an understanding of Cyrus' religious role in the ensuing doctrinal controversies within the Roman Empire, check out the relevant pages in Phil Booth's Crisis of Empire: Doctrine and Dissent at the End of Late Antiquity (2014). For a modern appraisal of Muhammad's life, it is worth reading Michael Cook's Muhammad (1983) and the more recent but perhaps more debatable interpretation put forward in Stephen Shoemaker's The Death of a Prophet: The End of Muhammad's Life and the Beginnings of Islam (2011). I know this is perhaps not the answer you are looking for, but I hope you find it interesting nonetheless!