This is a vey tricky to answer as there are numerous aspects related to medieval and early modern approaches to words like "emperor," and "empire."

First, the title of Emperor in a European context was definitely higher than king. Emperor comes from Imperator, as most, if not all, European peoples had at least some contact with the Roman Empire. *I will be summarizing a lot here, so if there is anyone else who wants to critique or provide greater insights that would be welcome!* After Christianization, Catholicization (in which I mean various other early Christian beliefs and sects were declared heretical and expunged from the main canon), and the spread of religious fervor across the Roman Empire, the Roman Empire became the main vehicle of Christianity though state-sponsored programs of church building and legal concessions made to churches. All of this tied into the creation of Rome as a central point of Christian religious belief, along with Jerusalem, Alexandria, and Antioch. The Pope, Roman (Christian) heritage, and connection to the spread of Christian beliefs through missionaries, created a pan-European vaguely Christian environment that was all made possible due to the Roman Empire.

But then the Roman Empire fell, or at least part of it did. The East continued to ebb and flow, but new threats were placed on the Pope: barbarian incursions, the Huns, Byzantine expansion, how to ensure security? The Pope needed a protector, and found one in the Franks to the north-west. He crowned Charlemagne Imperator Romanorum (Emperor of the Romans) in a mutually beneficial act that strengthened Charlemagne's legitimacy and access to religious support and provided security to the Pope. It is also interesting to note that Charlemagne abandoned the Romanorum part of his title shortly after his coronation possibly in order to maintain relative stability with the remains of the Roman Empire in the east. *Summary done.*

Okay, so what? What does all this Christianity, the Pope, the Roman Empire, and now the Holy Roman Empire have to do with emperor as a title above king?

Everything.

St. Jerome in the fourth century gave a provocative and powerful interpretation of the Book of Daniel, when the Prophet Daniel was asked by Nebuchadnezzar about his dream. The verses are 37 - 41:

"Your Majesty, you are the king of kings. The God of heaven has given you dominion and power and might and glory; in your hands he has placed all mankind and the beasts of the field and the birds in the sky. Wherever they live, he has made you ruler over them all. You are that head of gold.

After you, another kingdom will arise, inferior to yours. Next, a third kingdom, one of bronze, will rule over the whole earth. Finally, there will be a fourth kingdom, strong as iron—for iron breaks and smashes everything—and as iron breaks things to pieces, so it will crush and break all the others. Just as you saw that the feet and toes were partly of baked clay and partly of iron, so this will be a divided kingdom; yet it will have some of the strength of iron in it, even as you saw iron mixed with clay." 

What did Jerome have to say? This was a reference to four world monarchies that succeeded one another: Babylon (which Nebuchadnezzar ruled), Persia, Macedon, Rome. There can be no others, or else Daniel would have interpreted other kingdoms as the Prophet of God, which he did not. Therefore, these were four empires. Not just four, but the four empires of the world as stated in Daniel.[1] And relatively soon after St. Jerome's death, (western) Rome fell. The need for Pope Leo III to have a secure power as a guarantee for security provided a signature reason as to why Charlemagne was crowned Imperator Romanorum. But it wasn't until 1202 that the problem of making an emperor while the Eastern Empire was intact was resolved. Innocent III issued in his Venerabilem that Leo III had translated the Empire from Byzantium to the Germans with his crowning of Charlemagne in 800. This was known as the translatio imperii.

With the fourth empire transferred, none could be created. And indeed, in the sense of St. Jerome's interpretation and espousal of Catholic tradition, no new Empire was made. Instead, Peter Wilson notes that "Empire" was gradually changed to mean decidedly sovereign polities. It was not really until the 19th century that this changed, with the German Empire, Austrian Empire, Ottoman Empire, Russian Empire, Second French Empire, and Victoria as the "empress of India." For Peter Wilson, empire changed from a "world order" from the middle ages to a term for rulers to have for possessing a large state. [2]

After all, in the middle ages, was there an English Empire? Spanish Empire? French Empire? Danish Empire? Polish Empire? Note too, that often statements made by historians about the "pre-modern empires" refer to the one polity's settlements and control over large swathes of territory, in this case reflecting historical, sociological, and anthropological words and phrases such as imperialism. It is also important to note that once we reach the eras of colonization, conquest, worldly trading companies, etc. empire began to take a different meaning as explained in the previous paragraph.

There are also individual examples to consider that definitely limit and increase the complexity of what I have said so far -- was Spain in 1500 a Spanish Empire? What about in 1550, when ruled by Carlos/Charles/Karl V von Habsburg, emperor of the Holy Roman Empire? Did the British empire exist when fighting the American colonies? Or just when Victoria was made the "empress of India?" There are a lot of very important and reasonable limitations and holes to what I wrote above. And it could probably be debated for decades. But this does not limit the fact that the title of emperor was higher than that of king.

And very quickly, was emperor above, equal, or below the Pope? Depends on a lot of factors! To whom does this matter? To Luther at the end of his life, the Holy Roman Emperor was definitely more important! To Catholic patricians in Florence in the 15th or 16th centuries, the Pope would probably be more important as the German Emperor was partly (though significantly) responsible for the conflicts and wars with France in northern Italy during the Italian Wars. There was also a "Two Swords" belief -- that the Pope was the sword of the sacred and the Holy Roman Emperor the sword of the secular. In this case, they would be equal. But that did not stop repeated disputes over investiture, inheritance, payment, vassalage, etc. over centuries. Unfortunately, I have to say "it depends."

Also, definitive order of monarchical titles by power -- I assume you mean, emperor, king, duke, count, baron, knight (in descending order). In this case, no. Once again, it depends. Holdings of a ruler, loyalty and contracts with vassals, state of the realm, personal relations with neighbors, etc. all dictated relative "power" and the ability of one to enact one's will on the affairs of the realm. A rich Italian county or duchy in 1400 (recovered from the bubonic plague) with irrigation, connections to merchant ports for the purchase of goods, and a population to see to the maintenance and expansion of that realm's needs would make it, in many ways, potentially more powerful than the King of Scotland. But what does power here mean? As a powerful king before ideas of states developed would be one able to see to the needs of the realm and do his/her best to rectify ills, not declaring war for conquest and winning.

Source:

[1] Wilson, Peter H.. Heart of Europe (p. 38). Harvard University Press. Kindle Edition.

[2] Wilson, Peter H.. Heart of Europe (p. 163). Harvard University Press. Kindle Edition.