What causes the dotted line to appear in the green interpolated frames of my shape tween?
You either did something that contradicts the rules of a shape tween (that you have no groups and no symbols), or you don't have two keyframes. Remember that a tween is between two keyframes. You should check to make sure you have two keyframes and that in each you have no groups and no symbols.
My animation is acting funny. Shapes appear only in my keyframes, never in the interpolated frames. I know I did everything according to the rules because I'm not getting the dotted line. What's wrong?
Just because the dotted line isn't present doesn't mean you're doing everything correctly. For example, if you have some shapes in the start and the end of your tween, all will appear fine. However, if in either keyframe you have a group, text (which is like a group until it's broken apart), or a symbol, all these objects will disappear during the tween. The only time you see the dotted line is when all the objects onscreen are groups or symbols.
When I click in the green interpolated frames (Flash's territory), I can set the Properties panel Tween setting. I thought this area was under Flash's control. Why am I given access to it?
You're actually accessing the previous keyframe's properties. Remember that the span after one keyframe (before the next one) is controlled by that first keyframe. Any non-keyframes after a keyframe are under the influence of the first keyframe. With tweening, it's the same: Interpolated frames get drawn by Flash, but the previous keyframe controls exactly how the tweening will act. One keyframe controls its frame and all subsequent frames until the next keyframe. The good news is that the Properties panel gives you access to frame properties of keyframes without having to be really careful where you click. Just be aware of which keyframe you're accessing.