Is there a tool in Inkscape similar to the stroke Width Tool in Illustrator? In other words, is there a way to manipulate the width of a stroke by setting the weight at a given position or anchor point along the path?
Yes, there is the PowerStroke. In version 1.0, it will be possible to draw it with pressure sensitivity with the Pencil tool. Currently, the effect can be added via the 'Shape' option of the Pen/Pencil tool, or via the Path Effects dialog.