What is the paper cutting art called which does not involve folding?
This may simply be called papercutting!

From Papercutting on Wikipedia:


  Papercutting or paper cutting is the art of cutting paper designs. The
  art has evolved uniquely all over the world to adapt to different
  cultural styles.
  
  Silhouette can refer to the art of cutting outlines or portraits out
  of black paper. [...] Contemporary papercutting is also sometimes
  associated with the art of stenciling, itself being derived from
  techniques used in graffiti art. The use of hand-cut stencils in
  graffiti art has received international attention in recent years due
  in part to the artist Banksy.


There is also more information on Chinese paper cutting, although I can't verify the accuracy of it:


  "[L]ong before the paper was invented... people used thin materials,
  like leaves, silver foil, silk and even leather, to carve hollowed
  patterns for beauty. Later, when paper was invented, people realized
  that this material was easy to cut, store and discard. Thus paper
  became the major material for them to use, and people habitually
  called this artistry paper-cutting, or Jianzhi in Chinese."


And Jewish paper cutting:


  Jewish paper cutting is a traditional form of Jewish folk art made by
  cutting figures and sentences in paper or parchment. It is connected
  with various customs and ceremonies, and associated with holidays and
  family life.


A couple of examples: