Why is Satan commonly believed to rule in hell?
Some believe Satan is a Seraphim and chief yof the sinning angels, is probably the highest of all the angels. There are some who think that Lucifer was highest only among the rebel angels in hell.
This answer will be based on the Catholic traditions in regards to Satan and the nine choirs of angels.
Although it is now generally accepted that there are nine choirs of angels and each choir is of angel is different and ranked accordingly.
During the Middle Ages, many schemes were proposed, some drawing on and expanding on Pseudo-Dionysius, others suggesting completely different classifications. According to medieval Christian theologians, the angels are organized into several orders, or "Angelic Choirs.
Pseudo-Dionysius (On the Celestial Hierarchy) and Thomas Aquinas (Summa Theologica) drew on passages from the New Testament, specifically Ephesians 1:21 and Colossians 1:16, to develop a schema of three Hierarchies, Spheres or Triads of angels, with each Hierarchy containing three Orders or Choirs. Although both authors drew on the New Testament, the Biblical canon is relatively silent on the subject, and these hierarchies are considered less definitive than biblical material.
Choirs in medieval theology
St. Thomas Aquinas in Summa Theologica (1225–1274):
1.Seraphim, Cherubim, and Thrones;
2.Dominations, Virtues, and Powers;
3.Principalities, Archangels, and Angels. - Christian angelology (Wikipedia)
St. Thomas Aquinas' list is by far the most widely accepted list of angelic choirs and places the Seraphic order of angel as being at the highest level of angels followed by the Cherubim, and so on. According to St Thomas:
1.The angels that rebelled and became demons did not lose their nature or their connatural gifts. They cast away, by their sin, the grace in which they were created. They did not cast away the beatific vision, for they never had it. Now, if we think of angelic orders as orders of angels in glory, then, of course, there are no orders of bad angels. But if we consider angelic orders as order of angelic nature simply, there are orders among the demons.
2.Certainly, there is a precedence among bad angels; there is a subjection of some to others.
3.Demons of superior nature do not enlighten inferior demons; enlightenment here could only mean the manifestation of truth with reference to God, and the fallen angels have perversely and permanently turned away from God. But demons can speak to one another, that is, they can make known their thoughts to one another, that is, they can make known their thoughts to one another, for this ability belongs to the angelic nature which the demons retain.
4.The nearer creatures are to God the greater is their rule over other creatures. Therefore, the good angels rule and control the demons. - ORDERS AMONG THE FALLEN ANGELS
Thus stated it only makes sense that Satan was a Seraphim, for surely some of the Seraphim fell in great revolt against God. Although one can not say with certainty it is generally believed Satan was a Seraphim.
SIN OF THE FALLEN ANGELS
3.Lucifer who became Satan, leader of the fallen angels, wished to be as God. This prideful desire was not a wish to be equal to God, for Satan knew by his natural knowledge that equality of creature with creator is utterly impossible. Besides, no creature actually desires to destroy itself, even to become something greater. On this point man sometimes deceives himself by a trick of imagination; he imagines himself to be another and greater being, and yet it is himself that is somehow this other being. But an angel has no sense-faculty of imagination to abuse in this fashion. The angelic intellect, with its clear knowledge, makes such self-deception impossible. Lucifer knew that to be equal with God, he would have to be God, and he knew perfectly that this could not be. What he wanted was to be as God; he wished to be like God in a way not suited to his nature, such as to create things by his own power, or to achieve final beatitude without God's help, or to have command over others in a way proper to God alone.
4.Lucifer, chief of the sinning angels, was probably the highest of all the angels. But there are some who think that Lucifer was highest only among the rebel angels. - ANGELS: FROM THE TEACHINGS OF SAINT THOMAS AQUINAS