These results strike me as unsurprising. The effort here seems worthwhile, but analysis of the survey results leaves something to be desired.
First, about the overall question: "Thinking about the present situation, would you say that the world is becoming safer or more dangerous for the United States and the American people?"
There's nothing directly in the question about terrorism. Degrading relations and increasing tensions with North Korea and Iran are not primarily related to "terrorism."
A question about "winning the war on terror" presupposes that: 1) there is a war, beyond its declaration; 2) that "terror" is something against which you can wage war, as distinct from simply raging against it; 3) that "winning" such a war is necessary and important. To my view, none of these are obviously true.
The questions about public diplomacy are valid but not especially helpful. Failure at public diplomacy is not about declining to hire the right ad men. A successful public diplomacy is not even really about leadership; to be sure, a poor leader can screw up big time, but so can a buck private with a digital camera. Rather, achieving success in public diplomacy is the responsibility of the Public. That means You. Why is this seemingly so hard to understand?
I would propose for discussion the hypothesis that no major nation is doing a particularly good job at public diplomacy. Surely our public diplomacy is more effective than that of North Korea, or of Iran. When was the last time you met a North Korean at an industry conference? Have you ever talked with a North Korean about what it's like to live in the countryside? Iran is in somewhat better shape, but the contemporary popular understanding of Iran is dominated by the actions of its President -- to a much greater extent than for the United States, I'd argue.