The answer to pretty much everything you ask is "Yes."

One very important thing to keep in mind is that nowadays, we expect governments to have definite and internationally recognized borders, separate and well-defined of powers within a government, and a clear separation between church and state. But all of these concepts would not have been clear to medieval theologians or to medieval political thinkers. The political history of the middle ages is very much the history of normalization and codification of power.

Monarchies, aristocrats, cities, and religious institutions all existed in a space where powers interacted and overlapped. The Papacy certainly held that its authority anointed monarchs, but monarchs naturally had a different take on things (unless, of course, they could needed a Papal coronation to legitimize themselves). Bishops which theoretically represented the church in its various dioceses could be highly politicized figures performing a number of local government duties (and were often chosen from the local urban aristocracy). The Roman aristocracy used the Papacy's legitimacy to assert their authority over cities in Central Italy (and the Papacy was more than happy to humor them) but they could not extend this authority over the large mercantile cities of Tuscany, and had a complicated relationship with the Kings in Sicily and (later) Naples.

I wrote an answer on how the Papal States came to be here, which gives a lot more color on the topic. In that answer, I recount how the Papal States, while on the surface a unique theocracy, in reality represent a possible and objectively predictable path in state building: for various reasons, in the city of Rome the local bishop (the Pope) swelled in power while the the senate-council apparatus waned, instead of the other way 'round as happened in the cities of the rest of Europe. This coupled with the fact that in Italy's political development there emerged no central monarchial figure superimposed upon the urban Bishops which, in the Roman fashion, acted as joint civil and secular leaders in their communities (perhaps in part because of the presence of such an authoritative and central "Pope" figure in Rome) created a space for there to be a "Papal State," and also allowed the Bishop of Rome to ultimately appropriate authority over an institution that could very much be envisioned as the last remaining institution of the Roman Empire (or at least, that’s how the medieval Church liked to imagine itself operating: a hierarchical network of spiritual leaders bestowing legitimization over rulers and institutions, much as the various proconsuls has done in the time of the ancient Empire; with the reality, as is often the case, being much more nuanced).

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I'm not a historian, just an amateur enthusiast, so - apologies if this needs to get removed because it's not formatted correctly. Also if I get something wrong, I hope someone will correct me.

But since the OP here does not address any particular era in history, and since I am reading Peter H. Wilson's book Heart of Europe (an enormous and thorough history of the Holy Roman Empire), I will offer a comment. Its early chapters address the relationship between the papacy and the HRE - and later, by extension, the Papal States and the relationship of the papacy to secular powers/kingdoms. So, in case it helps:

The short answer is that the Papal States were landholdings of the Catholic Church itself. They were the temporal manifestation of the Church and its power, and were directly under its management, including in the day-to-day affairs. It is important to understand that the Roman Empire's "fall" took a long time, hundreds of years - and towards the end of what we tend to think of the Roman Empire's existence, it was largely Christian - Wilson writes:

After more than three centuries of persecuting Christians, Rome adopted Christianity as its sole, official religion in AD 391. This step partially desacralized the imperial office, since the singular Christian God would not tolerate a rival. The emperor no longer considered himself divine and had to accept the church's development as a separate institution throughout his empire. These changes were eased by the church's adoption of a clerical hierarchy modelled on Roman imperial infrastructure. Christian bishops resided in the chief towns, exercising spiritual jurisdictions (dioceses) that generally matched the political boundaries of the empire's provinces.

Note this is quite old, though, in 391 - before what the "Papal States" later came to be. But it sets the stage for the later development of the idea that the Church sees itself principally not as a temporal power, but as a parallel, spiritual power.

That being said, though, the Church also wanted its own lands, military forces, and internal infrastructure for self-defense. And - critically - the Popes towards the late 700s felt that since the Church's primary ambit was spiritual, it would need a secular protector.

That protector would be the Holy Roman Emperor - the first one being Charlemagne, or Charles I. Wilson offers an explanation of this "title" roughly as follows (I'm summarizing here): "Holy" because the emperor was seen as the secular/temporal protector of Christendom; "Roman" because Charlemagne had conquered a ton of territory that was formerly the Western Roman Empire; and an "Emperor" because the "mode" or "style" of rule was more "empire-like" than monolithically monarchical, with constituent territories under the rule of the Emperor allowed to maintain a degree of their own autonomy while on the "periphery," provided they obeyed the "center" when instructed to. (I am not doing justice to Wilson's account here, which is nuanced and good, and difficult to concisely summarize.)

Anyway, after ruthlessly consolidating his power over the Lombards, the Frankish king Charlemagne made a deal with Pope Leo III to form the HRE. Wilson writes:

Three aspects stand out. First, the Empire was a joint creation of Charlemagne and Leo III, 'one of the shiftiest occupants of the throne of St Peter'. Accused of perjury and adultery, Leo was unable to assert authority over the Roman clans, who orchestrated a mob which attacked him in April 799, nearly cutting out his eyes and tongue - acts of mutilation that were considered to render their victim unfit for office. Already at his accession, Leo had sent Charlemagne a banner and the keys to St. Peter's tomb, symbolically placing the papacy under Frankish protection.

...Just what Charlemagne thought he was getting into is not clear... it is more likely that Charlemagne saw his accession as a way to consolidate his hold over all Italy, since the former Lombard kingdom [which he had just conquered] only covered the north, whereas the idea of the Roman empire had a great resonance throughout the entire peninsula. Additionally, in accepting the various religious symbols, Charlemagne signalled his partnership with the pope as joint leaders of Christendom.

This meant that the Holy Roman Emperor and the Pope were not really "co-rulers" of Christendom, exactly - but partners in the protection of the Church, which was now an entity in its own right, and distinct entity from secular powers.

As for the actual operation of the Papal States themselves, Wilson is not really writing a history of them, so he does not address this in a focused or direct manner. However, he does address this obliquely. The papacy needed its own internal taxation, management, and organizational structure in order to maintain itself, and the Papal States become just that. But even from the beginning, it was complicated. Wilson:

...[Christian] opposition to secular authority made sense during Christianity's time of persecution by the Romans, and was underpinned by the doctrine of Christ's second coming, which suggested the secular world was of little significance. However, the delay in the Messiah's return made accommodation with secular authority unavoidable, as exemplified by St. Paul's response to the Romans: 'Every person must submit to the supreme authorities. There is no authority but by act of God, and the existing authorities are instituted by him; consequently anyone who rebels against authority is resisting a divine institution.' (Romans 13:1-2).

Christians owed obedience to all authority, but their duty to God trumped that to secular power. It provided impossible to agree whether they should suffer tyrants as a test of faith or were entitled to oppose them as 'ungodly' rulers. ...In short, Christian thought tried to distinguish separate spheres of regnum*,* denoting the political realm, and sacerdotium, for the spiritual world of the church.

...Pope Gelesius I used the influential metaphor of two swords, both provided by God. The church received the sword of spiritual authority (auctoritas), symbolizing responsibility for guiding humanity through divine grace to salvation. The state received the secular sword of power (potestas) to maintain order and provide the physical conditions to enable the church to perform its role. Christendom had two leaders. Both pope and emperor were considered essential to proper order. Neither could ignore the other without negating his own position.

...Clergy and laity generally worked together, while spiritual and secular authority generally proved mutually reinforcing rather than conflictual. Nonetheless, the issues remained clear enough. Secular power was inconceivable without reference to divine authority. Likewise, the clergy could not dispense with the material world, despite waves of enthusiasm for those seeking 'freedom' from earthly constraints as hermits or monks. The Franks gave Ravenna to the pope through the Donation of Pippin in 754, presenting this as restoring it to the Patrimonium, yet they retained secular jurisdiction over the entire area, asserting claims not dissimilar to those of the Lombards they had just displaced.

The "Two Swords Doctrine" came to inform a great deal of how the Church dealt with secular powers in general for hundreds of years. The Church's authority was spiritual, not temporal, and in that sense, it claimed spiritual authority over monarchs, the administrators of a city, etc. However, it did not claim temporal authority - the pope could declare that a king's behavior was pious or impious, but could not necessarily order that king to do something in particular with his tax revenues, for example.