Is there a shortcut / an app to center the mouse on the current window?
Keyboard Maestro can be setup to do this. I know this is a heavy weight solution for a simple task. However, if your already have KM or if you find value in it for automating other tasks,  it is a great option. 

Here is how to set it up:


Note: this is centered. The (0,0) is relative to the center of the window.