Every time I try to drag a rectangle I either get a red triangle, or the complete object is highlighed.  

In Inkscape, objects can be selected and nodes of paths can be selected.

This differs from paint programs like PS and GIMP where a selection rectangle can encompass an area.

One way to make a bitmap copy of an area is to create a rectangle object with no fill or stroke, then use Alt+B to copy and past a bitmap of the rectangle's region.