If I press the "include alpha-button" in the Mix-Color-Node, I get white borders. Otherwise i've got a perfect alpha but no blue background.
You do not need a mix node, Use an alpha-over node. Although blender shows you a border around the cube, it is actually doing what you want. Use the following node setup. You do not need a scale node per se, I only have it because the background image is larger than the cube render
If you were to export a render with the shown node setup and import it into another compositing application you would see that this is the correct method. The alpha is retained and the composite shows the default backdrop from when you rendered it out of blender, but you can easily change that since the alpha is retained.
The alpha channel as viewed in another 3d app.
The node setup