I'm sorry that you're stuck with me as the only respondent. The forum seems a bit dead. But I'll do my best.
What role does ministry have in church authority?
I'm not sure I understand your question. Are you asking what authority ministers (i.e. pastors) have over their congregations? You are right that it varies. The LCMS considers itself a "congregational" church. So, the members own and maintain the property; they manage all the "business" affairs. However, the pastor is considered a "called" minister. So, God put him in the pulpit, not the congregation. He, therefore, is the spiritual/moral leader of the congregation. He has authority over what is taught (preaching of the Gospel) and the administration of the sacraments as well as over things like marriages.
Do Lutherans believe any group of people anywhere can start a church whenever they want?
This again goes to the idea of being called. Yes, anyone can be called by God at any time in any place. But there are two calls. There is a call to serve God and a call to serve the Lutheran church. Only those called to serve the Lutheran church are recognized as pastors with the authority mentioned above. Being called does not mean one has been mystically endowed with knowledge. One still needs to be educated. So, if someone believes they have been called by God, a Lutheran would encourage them to attend a Lutheran institution of education that matches their call. That could be a seminary, but it could also be a Lutheran college for parochial teachers, etc.
A Lutheran won't deny that someone has been called by God. They just won't accept anything that differs from the Confessions as part of that call. Those who claim to be ministers and are outside the Lutheran church are viewed as members of human institutions. They can still be Christians and they can still serve God. They're just not recognized as pastors. So, a Lutheran pastor will not engage in "ecumenical" services, etc. They're more than willing to talk with anyone about Christian matters, and they're more than willing to acknowledge them as representatives of different groups of Christians. They just won't interact with them through preaching and the sacraments, nor give any impression that the Lutheran church endorses what they're doing.
I read somewhere that in Lutheranism, the Bible is the highest authority but not necessarily the only authority. The church can be an authority too. Can anyone expand on this further?
Scripture is the final authority. Again, however, reading the Bible doesn't somehow produce some kind of mystical understanding. Christians need guidance (Proverbs 1:1-6), and the Church is what provides that guidance. Therefore, Lutherans - even Lutheran pastors - will listen to someone who challenges them upon the basis of Scripture. However, the Confessions (the Book of Concord) is our exposition of what Scripture means. So, challenges that claim a new revelation differing from the exposition of the Confessions has little chance of being accepted.
As mentioned above, being Christian does not excuse one from learning. And learning comes from the wise within the Church. So, the interpretation of Scripture, though necessarily guided by the Holy Spirit, is not a mystical process apart from the Church. Rather, it takes place in the Church under the guidance of a teacher. And that teacher learned from a previous teacher in the same way. So, there is a very real and identifiable history of teachers who have subscribed to what the Confessions express.
Why is it that some nations have national Lutheran churches?
Well, the national churches you see in Europe are an unfortunate distortion of what Luther wanted. The modern nation is based on the principles of the Enlightenment, not Lutheranism, and that has caused a lot of problems. Even people in the U.S. who tout us as a "Christian" nation miss Luther's meaning.
But, with that said, in Luther's mind there is no such thing as a "secular" institution. Lutherans speak of the two kingdoms and of vocation. The Church is the Kingdom of the Right, and it's tools are the Gospel and the sacraments. That is how God meets us in the Church. Basically everything else is the Kingdom of the Left, and it is how God directs our lives. I am an engineer because God predestined me to be an engineer and to serve him through my vocation of engineering. This dichotomy can be viewed in many ways: the kingdoms of left & right, Law & Gospel, Sanctification & Justification, works & grace, virtue/morality & faith, etc.
So, government should be a Christian partner of the Church, and yet not connected to it. Government should be done via Christian principles, yet not controlled by the Church (nor the Church controlled by the government). Government officials need pastoral advice, and yet pastors should not hold government offices.
Hopefully that helps, but you'll probably have some follow-up questions, so ask away.