Why inversion of wave for rope fixed at one end? Setup: A rope is fixed at one end to a wall. You swing the other end up and down once. A wave starts travelling. It moves, hits the wall, then flattens, then is created again underneath (inversed), and then starts travelling back.

Question: The inversion itself makes sense. The wall responds with a force downwards on the rope when the wave in the rope tries to pull the wall upwards to continue (Newtons 3rd law). But why is there a delay?
At the point where the wave is perfectly flat, what is in that exact moment pulling the rope downwards? In that moment the wave is not pushing the wall upwards anymore, so the wall will also not give any force. Or am I wrong? Where to has the energy in the travelling wave been converted which gives it back to the rope in the flattened moment?
 A: 
At the point where the wave is perfectly flat, what is in that exact moment pulling the rope downwards?

Nothing. The force of the wall on the rope vanishes at that instant.

In that moment the wave is not pushing the wall upwards anymore, so the wall will also not give any force.

Right.

Where to has the energy in the travelling wave been converted which gives it back to the rope in the flattened moment?

It's in the kinetic energy of the rope.
A: Putting comment into answer, as requested:
At the point where the wave is perfectly flat, nothing pulls it downwards. But to continue moving it doesn't need any force - it just obeys Newton's first law and moves with nonzero velocity.
A: Think of it like this: when is the kinetic energy a maximum in this oscillation? At the point when $y=0$, or at the point at which the rope is flat. If there were a force pulling it, it would violate conservation of energy as from that point thereafter energy starts to transfer to potential energy of oscillation. 
In short, the energy is transferring between kinetic and potential energies in the system. It's worth noting also that the transverse movement is the only real "movement" in the wave so to speak; that is, no point on the string is actually moving horizontally if you think about it. 
