Say we have a metal bar in space aligned horizontally and we start rotating it counter-clockwise about its left end. Then, the sin of the angle from between the horizontal and the bar is the y coordinate of the far end divided by the length of the bar. So, I understand that sin/cos are related to projections of things when you rotate them. Is this the fundamental idea? Why do sin curves look the way they do? Why is the slope most negative when sin(x) = 0? I'm looking for an intuitive answer, not just that the derivative of sin is cos.
Yes, that's the idea. The end of the bar is moving at a constant speed, so the $y$ component of its velocity (which is the rate of change of $\sin(\theta)$) is greatest in magnitude when all of that velocity is in the $y$ direction, and that happens when $\sin(\theta) = 0$.
Graphically you can see that at $\theta = 0$ the derivative of the $\sin$ (the $y$-value) is changing the most as opposed to anywhere else (other than the other side of the circle at $\theta = \pi$):
Here's another picture which shows that the shape of the sinusoid is due to the definition of sine in the first quadrant (forgive the approximate nature of this sketch--but it is basically correct):
I think the other answers fully answer the question, but this might be helpful. My favorite real-life visualization of a sin curve comes from a telephone handset cable:
When stretched out on a flat surface, and viewed from above, the shape of the cable approximates a sin curve. The equations for the center of the cable will be something like:
where R is the radius of the cable spiral, and A depends on how much the cable is stretched.