If the United States disappeared, how would the world be affected? Which nations rely the most on American support, patronage, partnerships, trade, and military protection?

Other than the sudden drowning of some 315-odd million people, lots of bad things that would probably kill tens of millions of more people. First off, the planet itself would hardly notice. The total mass of the US is [math]2.28*10^{16}[/math] kg†. If we go further and axe

Should more American troops be stationed in Europe?

As a Russian living in Lithuania i'd say damn YES. We need those American troops to protect us. Lithuanian military is as good as nonexisting, and EU has little resources, very little will, and no legal mechanisms to protect us. It is only the U.S. who is capable of doing so. Probably other EU countries except the Baltics could