I don't know what the tool is actually called since the name changes with each new tool (polygon, donut, etc.). Rather than a solid or textured fill, I'd like to drop an image into an irregular shape.
Masking the image with the shape would be one way to accomplish this.
Place the shape above the mask, select both, and choose Modify > Mask > Group as Mask. Now, within the Layers panel, you'll see that the two objects occupy a single layer. Click on the mask thumbnail, and you'll see the mask's properties in the Properties panel. You can choose between a "Path outline" or "Grayscale" mask, and select whether or not to "Show fill and stroke".
Yes, I think so. You could either create a series of masked objects—one shape per bitmap, each grouped as a mask—or you could group all the vectors (Modify > Group) and then apply that group as a mask to all the bitmaps. When using Group as Mask, the topmost object in your selection is applied as a mask to all the underlying objects in your selection.
The options or properties will vary depending on what type of object you're using as a mask. If the options are "alpha channel" or "grayscale", then you must be using a bitmap as the masking object; if you were using a vector as the mask, the options would be "path outline" or "grayscale", and "show fill and stroke" would be selectable if the path outline option were chosen.