1) What was the only Roman institution to survive the fall of the Western Empire? Why was it so important? How did the Holy Roman Empire represent the power of this institution?

The institution that survived the fall of the last remnant of the Roman empire was naturally the church, which has even survived even until modern times.

It was important because it provided order, security, and stability in times when revolts and wars over power came one after another, what is more, it brought identity to a civilization who had lost it centuries ago

It is still important, since most of what it is known about roman civilization survived to our days along with the roman religion, including but not limited to the latin language which otherwise would be lost forever due to the continued break up of the once unified roman empire as well as the mixture of barbaric tribes and their customs which cause population not to understand latin anymore and eventually leading to the development of romance languages. Latin continued to be used by the church nonetheless.

The Holy Roman Empire represented such power by spreading this identity as far as they could and assimilating the emperor and the rest of institutions under his grasp. In this way the roman catholic church melded in a church - state alliance where church was for long time an entity as big and powerful as the state itself. This is how the hierarchy and ways of the roman church exist and continue to influence many people around the world today.