Why did the concept of "divine rights of the kings" is thought to begin in the middle age Europe, and not in Christian Roman Empire? Did Roman Emperors have no divine authority?

It did, it all goes back to the "render unto Caesar" in the Bible. You could say Pope Gelasius I was actually trying to reduce the respect paid to the Emperor with his Two Swords theory, as it was too high and eclipsing that of the church.

That the proper and divinely-ordinated ruler of Christians must be a Roman Emperor lived on in the Byzantine Empire and in Charlemagne's, later on in the Holy German Roman Empire. The idea was that kings - rex in singular, reges in plural - are more or less employees or subordinates of emperors.

In fact, one way to understand a rex is that it was a military rank in the Roman armed forces. If they would hire auxiliary troops from foreigners, which would often mean the chieftain or "barbarian king" of those people also leading them as troops, they would give the boss the military title or rank rex, roughly general.

(Sauce: I got this idea from Hilaire Belloc's Europe and Faith which is not fully accepted amongst historians and debated, but this is at least one of the viewpoints. Also see the rex-rix clusterfuck below.)

This leads to these lovely misunderstandings that "Alaric, King of Visigoths, sacked Rome". In fact saying something like a Roman general of Gothic origin with the Rex Gothorum rank/title led troops of mixed ethnic origin in a typical Roman civil war after he was not paid to sack Rome.

One thing that complicates this whole set of issues is the rex/rix/reiks confusion. Maybe it would be better to say the rix, not the rex was a Roman military rank and Alaric was a rix. But whether the medieval kinghood comes from rex or rix is difficult to say. Since the rix is understood both a as a leader of troops and a "lesser king", like a basiliskos, under the Emperor, and some of the biggest bosses e.g. Barbarossa for called emperors for a long time, anyway, you can see how the concept of rex-rix king as a lesser ruler under an emperor, something sort of a general, develops.

For example, when King Saint Stephen of Hungary took Christianity in year 1000, bordering two Christian Emperors, was generally expected to be subordinate to at least one of them. One way to wiggle out of this was to try to make up an independent divine justification for his kingdom, something that would look about as sacred as the divine rights of emperors, and tradition/legend says he figured the real queen of Hungary is Virgin Mary, Regnum Marianum, her governor is the Holy Crown and the king merely the representative.

At any rate, the Middle Ages can be seen as kings wiggling out under from Emperors, France, England, Hungary etc., and finding independent divine justifications for their rule.

Another absolutely core concept we need to look at here is the translatio imperii as in how could kings claim to be in succession of emperors, being their heirs, representing their divine rights. Imperium was a sacred concept, and kept far, far alive into the Middle Ages and on, as kings wielding imperium translated from emperors. Understanding Imperium is absolutely core to understanding the political philosophy of the age.