Good questions.

So the root meaning is basically "dominion" which ties back to Genesis.

As for your second question about coercion, that is a bit more complicated.

The simple answer is that there are plenty of just ways a person may attain authority and rule. That someone would attain authority isn't in itself wrong but, again, an inevitability.

I also should say that coercion once someone is IN authority is baked into the pie. It is not a matter of if they will be coercive but how they will be coercive. Or, put another way, to what end.

For example, I am coercive with my children—I will discipline them if they are rebellious to my authority—but my coercion is aimed in the direction of them being self-disciplined, respectful, and responsible individuals who are not mastered by their emotions and impulses and sins but rather learn how to submit themselves in righteousness to God. My authority as a father is to love them enough to teach them to obey all that God has commanded. That involves coercion.

On the flip side, If I were a tyrannical father I would still have authority over them because of my "governing office" as father but my rule would be using coercion for MY OWN gain, not my children's gain.

This, in principle, is true of magisterial rulers. God calls rulers to love their people and place. To use coercion in righteous ways for the benefit and flourishing of the people.

You specifically asked about attaining power through coercive means. On that note, and this is a harder reality to grapple with, God allows, for his purposes, selfish and evil men to do such things. Just as men can come to power through just means, so they can through unjust means. God is infinitely wise and we often do not understand why he allows all that he does. But, as was stated before, sometimes it is to discipline an unruly peoples. To discipline and rebuke in order to call them back into righteous submission to Christ.

Pardon the novel I wrote in response here. Does all that make sense?