Which Christian churches have an entirely unpaid/lay clergy? Of those Churches what is their form of governance that allows them to run smoothly lacking fulltime ecclesiastical professionals? How does one ascend into a leadership role in the Church?
Quakers and Mormons have no paid staff in their local assemblies.  In the national bodies, however, there is usually some renumeration or support, in order to retain staff.  The truth is that very few would consider it wise to have a national organization without any accountability, which is what an all-volunteer lobby would be.

Additionally, many more Baptists than you might think have volunteer staff. Finally, there are a great number of unpaid missionaries, itinerant preachers, and supporters of various causes that may fit into this category. Usually the local pastor is supported by the congregation, but the national staff has many more volunteers.

Finally, if you look at what a Roman Catholic priest earns, it's pretty close to nothing.

All that said, most mainstream Protestant churches will pay their clergy, based on Paul's commentary on paying preachers in 1 Timothy 5:18.  Those churches relying on unpaid staff tend to be less mainstream and more "feeling" driven, with the feeling being that preaching is a gift like any other.