I recently wrote an answer to this question that touches on some of your questions that I will repost here (also pinging u/Zeegisdik , who also wrote on that question). I can try to address followup questions here.

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The run of Germans in the 11th Century and the French popes in the 13th and 14th reflect political influence by secular powers in those periods. And they were both, in a way, related ultimately to the six centuries of Italians, interrupted only by the elections of the Spaniards Callixtus III and his nephew Alexander VI (both members of the influential Borgia family) in the 15th century, and by the Dutch Adrian VI, who reigned for less than two years near the beginning of the Reformation. Get ready, because this isn’t going to be a short answer. We’re going to cover, to the best of my ability, the history of papal elections and the pope’s relationship with secular rulers for the past thousand years or so. Strap in.

Today, when the pope dies or resigns there is a highly formalized process for choosing his successor. The College of Cardinals are the highest ranking officials of the Church hierarchy. Even though many of them are posted in various positions all over the world, each of them is appointed to be the titular head of a church in the city of Rome. This officially makes them Roman clergy, thereby eligible to choose the Bishop of Rome. They lock themselves into the Sistine Chapel, where they discuss, pray, and vote by secret ballot. The process takes many rounds, often stretching several days in modern times, because the election is decided by a two thirds majority. If you were around to watch the news coverage during one of the more recent conclaves in 2005 or 2013, you may recall that the paper ballots are burned after each round of voting. The smoke coming up the chimney is supposed to be colored black if the vote resulted in no winner, and white to announce that a new pope has been chosen.

Such a process was not in place during the first thousand years of Christianity. We know precious little about how the earliest Bishops of Rome were chosen. It appears that, at least in some cases, both clergy and lay people (including emperors and kings) were involved in papal elections. Byzantine Emperors were involved for a time, as were Frankish kings. The popes were technically elected, but a powerful ruler might “nominate” a preferred candidate, or communicate their displeasure with an objectionable one. Beyond that, this period isn’t my speciality. What is clear is that as the papacy gradually became a more powerful institution, secular rulers sought to influence both who was chosen and how they ruled. The counts of Tusculum, powerful Italian nobles, wielded enormous influence over the papacy beginning in the early 10th century. They even managed to have three members of their family chosen as popes in the 11th century. The Holy Roman Emperor weighed in as well, essentially appointing a number of popes (thus the run of Germans, but also remember that the Holy Roman Empire of the High Middle Ages was much larger than Germany is today, and plenty of "Italian" popes were technically born in the empire).

Incidentally, if the idea of a secular ruler influencing papal elections sounds old fashioned and corrupt, you should be aware that a number of Catholic rulers claimed the right to veto an unacceptable candidate into the 20th century. Emperor Franz Joseph of Austria successfully vetoed Cardinal Rampolla, the leading candidate at the 1903 conclave, and Francisco Franco tried (but failed) to block the choice of Cardinal Montini as Pope Paul VI in 1963.

But I digress. The Investiture Controversy of the 11th century marked a new phase in the relationship between the pope and secular rulers. It's a bit before my area of specialization, so I don't feel comfortable getting too deeply into it, but suffice to say it began as a dispute between the pope and the Holy Roman Emperor over who had the right to appoint (or “invest”) church officials in the empire. The emperor had exercised this power in practice previously, but the pope claimed authority. The disagreement eventually evolved into the basic question of who held the higher power: the Church or the secular authorities. After decades of war in Germany, the pope and his supporters (mostly) won out. The pope had the right to appoint high church officials (bishops, archbishops, etc.), though the emperor and other secular powers had authority in purely secular matters.

In the aftermath of the Investiture Controversy, the pope’s power was near its peak. Urban II was powerful enough to call for a crusade of all Christians against the Muslims in 1095. But more importantly for this discussion, the emperor’s influence in choosing the next pope was greatly reduced. The papal bull In Nomine Domini of 1059 firmly established that the College of Cardinals was the sole electoral body responsible for choosing a new pope. That was before the Controversy really heated up, but the emperor’s failure to retain his privilege of investiture even in his own lands meant that he had little ability to force his will in Rome. The rules were further tightened in the 13th century, requiring the cardinals to remain secluded together until a new pope was chosen (this following a nearly three year interregnum from 1268-1271 during which three out of the original twenty cardinals died, the cardinals were given only bread and water, and ultimately the roof of the building where they were meeting was removed, before they finally came to a decision and elected Gregory X).

Thank you so much for the answer. This definitely filled in a lot of holes for me. Much appreciated.