"Did not Jesus recognize the role of state when he said 'Render unto Caesar…'? Just asking not defending." [No fallacy alleged in this particular, but it will be answered below.]
"Romans 13 teaches that all governments are godly, good, and must be obeyed! The passage is speaking about the Roman government. The book is titled Romans. Written by Paul, a Roman citizen to the Christian church in Rome." (Latter part, Kermit Hauge)
Regarding, "Render unto Caesar"; in full (Matthew 22:17-21):
Tell us therefore, What thinkest thou? Is it lawful to give tribute unto Caesar, or not? 18 But Jesus perceived their wickedness, and said, Why tempt ye me, ye hypocrites? 19 Shew me the tribute money. And they brought unto him a penny. 20 And he saith unto them, Whose is this image and superscription? 21 They say unto him, Caesar's. Then saith he unto them, Render therefore unto Caesar the things which are Caesar's; and unto God the things that are God's.
Jesus gave an answer to satisfy both parties, while pointing out that they owed Caesar nothing. If he had meant, "Give the thugs what they demand", he would have said so; the words existed. Even if he had said that, it would not have made tax collecting thugs moral. And it is extrapolation to assume "Caesar" means "any state"; although the same nothing is owed. Remember, Jesus's message was not (to the disappointment of many) political, but spiritual—the salvation of souls, not (yet) the millennial kingdom or even the driving out of the Roman occupiers.
Regarding Romans 13, the passage is speaking about a hypothetical authority that (1) is of God, and (2) is not a terror to good works. That eliminates any modern state right off the bat; they are the "evil" spoken of, and not any power ordained of God any more than the thug that breaks into a home and holds the family hostage is a "power ordained of God" because he has control (authority—same word as power in the passage) over them.
The passage is of course not speaking about the Roman government merely because Paul wrote a letter to the Romans. Paul was intimately familiar with the Roman government, and would not have called them "ministers of God for good". Jesus, his disciples, and Paul himself all disobeyed governments and religious authorities to preach the word, and to preserve their own lives. The idea of God disobeying himself is something called a "contradiction", and means that a premise is wrong: in this case, that the passage is saying that all earthly governments must be obeyed.
Since God hasn't established any of the existing states, they do not have authority—just force.
Continue the passage; see the references about "ministers for good", only the evil need fear, etc. It clearly doesn't fit any known state then or today; rather, the passage characterizes an ideal godly authority that can be called God's servant. If Paul were calling evil states good, he would be going against Isaiah's admonition, "Woe unto them that call evil good, and good evil", which, as a rabbi, he would have known well.
An ignorant reading requires people submit to home invaders to do whatever they want—theft, rape, assault, assault of others even—just because they have temporary power, and not resist, attempt to escape, etc. It's monstrous, yet there exist so-called Christians that would justify such things. (DBR)
Romans was written by Paul while on his way to Rome and prison. According to many Bible scholars and language experts, both believing and non believing, Romans 13 was a way to tell fellow Christians that the only Ruler they had was God and Him personified by the risen Christ. He knew he still had work to do once he got to Rome and couldn't come straight out and tell the Romans he was writing the epistle to, to rebel openly against Rome. Reread it, the version you are accepting as fact is the same reading of it that Hitler commanded church leaders to read to their congregations, with the exact same explanation. We as Christians are not to back a government or authority who kills for no reason.
The earliest Christians broke the law every time they met; that right there puts lie to Romans 13 telling us to roll over and obey the authorities. (Eileen Jerome)
Related: Jesus is an Anarchist.