The early part of chapter thirteen would not have gone over well in the ancient world (any more than it does in our own day). Rome was not a democracy – at least not in the sense we think of it. Its emperor was a profligate but he was not the only one. Corruption was rife. Oppression was brutal – especially of the underclass. Why would God command the support of such an unworthy entity?
It has to do with respect for established authority. Government is there to protect the innocent and punish the guilty. It may not do its job well, or at all, but that’s its job. It is not under the oversight of the ruled, but the oversight of God and (despite American views to the contrary), answerable really only to Him. Because government is a full-time servant of God, it deserves to be supported (with taxes, revenues etc.). Christians must respect and obey governmental rule unless it conflicts with the revealed will of God but even then, civil disobedience is limited to the issue at hand. For example, just because the government protects a womans’s “right to choose” or a same-sex couple’s right to marriage, conflicting with God’s will, does not give Christians the right to overthrow the government or refuse to pay their taxes. The government is answerable ultimately and solely to God.
The matter of government sovereignty was highlighted recently in the 2013 governmental shut-down. Privately owned businesses situated on government land had to close. Why? Because though the building might be privately owned, the land on which it sits is the sovereign property of the United States which has total authority. One may not even sue our government without the government’s express permission.
Christian people were often accused of crimes against Rome. Even Jesus was accused. But they were never guilty. Christians, as far as it is possible, must be seen as law-abiding, respectful citizens.