calculus and analysis – Is it possible to ask Mathematica to separate all the terms containing $x$ and $y$ in the given function?

You may first expand your expression. Then you may collect all terms containing x/y and all terms not containing x/y.

Now, in your example, every term contains x because of the prefactor: E^(I (t (v - 2 w) + x)). I therefore delete the x in the prefactor to get a more telling example:

f(x_, y_, z_, t_, w_, v_) := 
  E^(I (t (v - 2 w))) (2 E^(I (3 t v + y)) + 16 E^(I (2 t w + y)) + 
     2 E^(2 I t w) t^4 + 3 E^(I t (v + w)) t^4 + 2 E^(4 I t w) t^5 - 
     2 E^(I t (v + 3 w)) t^5 + E^(I (2 t v + y)) t^6 + 
     8 E^(I (3 t v + 2 t w + y)) t^6);

Now we assemble the terms with and without x/y:

fwith = Select(Expand@f(x, y, z, t, w, v), FreeQ(#, x | y) &) // Simplify
fwithout = Select(Expand@f(x, y, z, t, w, v), ! FreeQ(#, x | y) &) // Simplify

And finally, if needed, you may redefine the original function:

fnew(x_, y_, z_, t_, w_, v_) := fwith + fwithout
fnew(x, y, z, t, w, v)