Many great European powers had been exploiting China's weak economic system to enforce their own trade regulations for years. The United States simply wanted to join the fray, so the Open Door notes were sent to the European powers basically stating, "Hey, you guys can't own all of China, make sure that all countries get to trade equally with them."
Just as a side note, the Open Door policy with China was extremely hypocritical for the United States, as the Monroe Doctrine of 1823 basically stated that none of the European powers could interfere with the Western Hemisphere... While no one was allowed to touch the Americas, the United States was apparently allowed to mess with the rest of the world.
"He built a second capital in the East, which led to the division of the Empire" would be the best way to describe how Diocletian affected the Roman Empire, since it had grown to a state where managing it under a single entity was too difficult.
