How is it possible to drag and select a quadrilateral/irregular are for cut and copy,
the way non vector graphics programs do? Where is the "lassoo" icon for the cursor in inkscape?

Raster and vector are two completely different concepts, you'd benefit from learning the differences and how it influences the approach.

Try looking into Clip and see if that is what you'd like to achieve http://en.flossmanuals.net/Inkscape/ClipAndMask