Does the European concept of “the balance of power” still have meaning?

The concept of "balance of power" from 16th and 17th century Europe does not apply today as it did then. In 16th and 17th century Europe, nation-states made and broke alliances that gave more weight to their value as a population/entity among other states. A small state could be "needed" in order to upset the balance of military threat for one side or another, because the resources that state offered might turn a war. Today, on the other hand, it only takes a few nukes to decimate a country. No one wants their country or people destroyed by war, but in 16th and 17th century Europe the destruction happened more slowly. It was possible to throw in the flag before you lost it all. Whichever side (alliance) could rip the most up fastest typically won. A terrorist sect with nuclear capabilities could bring any country on present day earth to its knees. You don't need fancy alliances to upset the balance. Just things that go boom.
As to whether the United States needing a counter-balance, the last time that happened was the Cold War. The balances achieved by 16th and 17th century European alliances were based on military threat. That is not the kind of balance needed today. Economically, competition (trying to keep up with the other side) is healthy, but the US is currently out balanced in that area anyway.

Prompt 5

There most definitely are patterns in the rise and decline of world powers. I think the largest factor is the dynamic fluctuations in human ethics and values, and the changes in human lifestyle that must accompany said fluctuations. Governments and religions are attempts to set ethical guidelines that the majority of people already abide by to improve the overall quality of life of humans(i.e. not killing others, not stealing, monogamous relationships). These values are guidelines designed to allow humans to function more efficiently as a group, and thus combine efforts more efficiently for the benefit of more people. The problem occurs when the opinion of the majority of people moves away from the guidelines of the government, or the practices of the government violate the ethics of the majority. Then stagnation and decline occurs, as the government/religion becomes more and more useless to the majority of people. The actions of the people begin to be guided less and less by the guidelines set by the government and more and more by their own, self-motivated, values and ethics. Eventually, when virtually no one follows the guidelines, the establishment collapses and the process of building a government/religion that is most beneficial to the majority begins again.
In summary, governments and religions rely on the support of people, so they represent values that people support. Thus, the people will uphold the establishment (picture a lot of people holding up a large building). But, with time, the values of people change (people holding the building are walking). If governments/religions attempt to uphold the same (old) values, slowly people holding the structure of the establishment up will leave (people walking begin to walk out from under the structure). With too little support, eventually the establishment collapses (the very few people who didn't walk fast enough: their knees buckle under the weight). Often a lot of the people still holding it up are killed (SQUISH).