Since the beginning of the worship of God, man has been involved in something akin to politics. Whether it was organizing a clan, clamoring for power over other members, or in more modern times participating in a sophisticated version with legislative bodies and such, politics is a natural state of social interaction. And since all men (and women) were created in the image of God, those who gain power are authorities under God to govern. Otherwise we would have total chaos and ultimately anarchy. The Apostle Paul understood this well and the first fourteen verses of the cited chapter of Romans deals directly with this, written at a time when the new and growing church was under the yoke of Rome.
The answer to the question herein posed is that no, God does not pick sides in elections. He gives us with our free will the opportunity to pick those who would govern us on earth, expecting us to understand the appropriate time and place for governance from man on earth and from God on greater things, things of an everlasting nature. Each election finds many candidates with widely varying views vying for office and God expects us to be smart enough to select those best suited by skill and good moral judgment to lead us. And we, in turn, have to live with the results of our choices, even if sometimes the difference between victory and defeat of the candidates is razor thin.
Having said that, the victorious candidates, once again men and women who are created in God's image, are expected to govern in a godly fashion. It only makes sense that a creature of God is expected to live within the expectations of Almighty God who created them. And this is where the problems come in, for when leaders don't live up to their obligations God then finds displeasure and he expects us, the subjects, to devise ways to insure that the leaders face their shortcomings and find their way. It is for this reason that such procedures as impeachment and recall from office, even the oft mentioned Article 5 Constitutional Convention were designed by our early leaders to offer solutions, not through chaos but through orderly procedure. But man being man, an imperfect and sinful body, often falls short of this. And if that weakness turns us from a situation where the appropriate concerns of man by leaders aren't addressed by the proper governing laws of this earth and turn to something dark and sinister, that is when it is necessary to pray for God's divine guidance and intervention if necessary.
So when we go to the polls in all of this year's elections, think things through, ask God to guide you with His love and the code for living that He has given us, The Ten Commandments, to make wise choices. If we use our free will correctly and follow it, we will make the right choices, but it's up to us. God is, however, always standing by to guide us in times of strife, but we do have to ask for his help.
Dear Lord: We thank you for the gift of free will and being given the ability to determine our own destiny through this freedom. Help us to use it wisely and to ask you for guidance in these endeavors always. In Jesus' name we pray, Amen.