Question 0. Let's put the issue to Mathematica:

  D[x Cos[xy],x]
  
  D[xCos[x*y],x]
  
  D[x Cos[x y],x]
  
  D[x Cos(x y),x]

Out[1]=

  Cos[xy]

  General::spell1: 

     Possible spelling error: new symbol name "xCos"

       is similar to existing symbol "Cos".

Out[1]=

  y xCos'[x y]

Out[2]=

  Cos[x y] - x y Sin[x y]

  Syntax::bktwrn: 

     Warning: "Cos(x y)" should probably be "Cos[x y]".

Out[2]=

  2 Cos x y

Only the third one of these is right.

Up to Test 3 solutions