HI!

So I hope I can answer the areas of your question that I'm more versed in! Sadly I don't know a great deal on the Great Schism, at least not enough to want to go into detail on.

But on the matter on what extent of legality did Pope Leo III have to crown Charlemagne, we can pose the question in terms of legitimacy. To look at this we need to look back to where the papacy even derives all its claimed legitimacy from. Because historically the papacy has based all of its authority on the first Bishop of Rome who we know as Saint Peter from the Bible, one of the 12 apostles of Jesus.

It was Bishop of Rome, Callistus I (218-222), that was among the first to reckon that Saint Peter was actually the first Bishop of Rome. Then came Stephen I (254-257) who was the first Bishop of Rome to claim primacy of Rome based on Peter. Now this is in particular is very important because it is at this moment in history that actual scripture from the Bible is used to justify the claim of papal primacy, or the supreme authority of the Pope in all things Church related.

Gospel of Matthew 16: 18-19:

And I say to thee: That thou art Peter; and upon this rock i will build my church, and the gates of hell shall not prevail against it. And I will give to thee the the keys of heaven. And whatsoever thou shalt loose on Earth, it shall be loosed also in heaven.

By tying their "lineage" and office to Peter, the papacy was drawing upon Peter's own God given authority of the church. By claiming that the Roman church, overseen by the Bishops of Rome, was a continuation of Peter they now were able to draw upon all the legitimacy and authority on all things divine and sacred that Peter had once had the right to hold. This is then the basis in which Pope Gelasius I (492-496) formulated his Two-Sword Theory:

There are two powers, august Emperor, by which this world is chiefly ruled, namely, the sacred authority of the priests and the royal power. Of these that of the priests is the more weighty, since they have to render an account for even the kings of men in the divine judgment. You are also aware, dear son that while you are permitted honorably to rule over human kind, yet in things divine you bow your head humbly before the leaders of the clergy and await from their hands the means of your salvation. In the reception and proper disposition of the heavenly mysteries you recognize that you should be subordinate rather that superior to religious order, and that in these matters you depend on their judgment rather than wish to force them to follow your will.

Now, what Pope Gelasius I was trying to put forth here with his two swords theory was that there were essentially two powers at play on Earth. One being temporal power, in which kings held, and the other being spiritual power, in which the papacy held in supreme. In Gelasius' own quote it is clear which of these powers he sees as the more greater. He plainly puts out that by wielding spiritual power the clergy, or in our case the office of the papacy, wielded the bulk of the authority. That even kings and emperors who may rule over mankind should still be "subordinate" and "depend" on the judgment of the leaders of the church. Therefore setting the stage for an office wherein the Pope sat atop even the rule of Kings. And all this was only possible on behalf of the legitimacy sought through Saint Peter.

Now that we've set the stage for a world where the church of Rome was even above earthly kings, we can start building the foundations in which Charlemagne was crowned. To do this we must look back on Charlemagne's own lineage within the Carolingian dynasty, particularly his father the first of the Carolingian kings. Throughout the majority of the Merovingian dynasty's reign, it was the office of the mayor of the palace, the Carolingian's, who truly held power. The Merovingian kings who sat on the throne were merely puppets in a kingdom run by the Carolingian household. And to fast forward this story to the point; by 751 the Merovingian king Childeric III was deposed and sent to live the rest of his life in a monastery. This left Pepin III, the current Carolignian mayor of the palace, in perpetual control of the kingdom. Both in power, which he had already held, but also now in reigning royal image. But here is where the problem arises. What legitimate right did the Carolingian household have to the seat of rule in the Frankish Kingdom? Pepin saw this question as a situation as well. And before the deposition of King Childeric, in a bid to solve this pressing matter, he sent envoys to Italy to send a message to the pope at the time, who was Zachary I. In this message Pepin basically asked the pope if it was right for the one with all the power, himself, to not also hold the royal title of king.

Pepin decided circa 750 that he no longer wanted Childeric III to rule in his stead as the last of the Merovingian puppet kings. Pepin ingeniously used the Church to grant him his wish. He sent two of his most prominent missi, Abbot Fulrad of St. Denis and Bishop Burchard of Wurzburg to Pope Zacharias. They were instructed to question the Pope ‘“concerning the kings in Francia who had no royal power, and whether this was fitting or not.’ Zacharias responded that such a situation offended the divine order and urged that Pepin be made king.” 16 So in November of 751, the assembly chose Pepin as their king and St. Boniface, acting as papal legate, anointed him in the tradition of the ancient Old Testament Kings. This was an extremely important move, loosely allying the Frankish kingdom with the Church and solidifying Pepin’s power ^(1)

So it is here we see the first act of apostolic authority being used by a Pope to crown a secular ruler as king. This move was only later cemented by the official and personal coronation of Pepin by Pope Stephen II. But this was only done after Pepin took in the realm of Rome and its surrounding lands under his protection. Because it was around this time of the papacy of Stephen II that the Lombard kingdom was threatening the city and neighboring lands of Rome. And in a bid to save the city, Stephen saw it fit to attach his fate to that of Pepin's. A classic quid pro quo. Pepin held actual power, enough to subdue the Lombards and protect Rome. While the papacy held spiritual legitimacy, enough to set in the legitimate reign of Pepin III and his lineage.

It is here we can finally get into Charlemagne and his own coronation. But a differing coronation to imperial status. It is fitting that Charlemagne's own papal coronation draws parallel to the one of his father, Pepin III. For it was because of Charlemagne's duty to protect the pope and Rome that culminated in his coronation of Emperor. This was all due to the act of a few, lets say rebels, who went out and attacked Pope Leo III in Rome. Pope Leo in turn sought out the assistance of King Charlemagne to bring order back to the city. And upon arriving to the Rome and setting it straight, on Christmas day of the year 800 Charlemagne was crowned Emperor of the Romans. This account was witnessed and recorded down by Einhard, a palace official of Charlemagne, in his telling The Life of Charlemagne.

The reasons for his last visit were not just there, but rather because the residents of Rome had inflicted many injuries on Pope Leo, putting out his eyes and cutting off his tongue. This forced him to appeal to the loyalty of the king. Thus Charles traveled to Rome to restore the state of the Church which was extremely disturbed...It was at this time that he received the title of Emperor and Augustus. (Pg. 38)

So grateful for Charlemagne's help in capturing and subduing the dissidents who would harm him, Pope Leo decided to crown Charlemagne Emperor. Pope Leo had strong precedence to back him up as we have seen. From the early claims of Pope Callistus I that Peter was the first Bishop of Rome. Which was then affirmed later on by Pope Stephen I by use of written scripture that essentially established the primacy of Rome based on Saint Peter. And all of this culminated in perhaps the most important doctrine put forth by Pope Gelasius I, that established the papacy and religious order above that of even secular rulers. All of these early claims and doctrines set by the Popes helped make possible the setting of the stage in which papal authority would then be able to be used in such a way where the crowing of kings and even emperors was seen as legitimate and even proper.