If I don't care about dim light or darkness, can I make an LCD-display sunlight readable by removing it's backlight and replace it by a mirroring surface, followed by a diffusor?
In theory, yes. Fundamentally the only difference between a backlit display and a reflective display is that the former has a light at the back and the latter has a reflective surface at the back. Of course, performance will be variable, depending on how the rest of the layers stack up.
But it's certainly possibly to get the behaviour, if not performance, of the various modes (reflective, backlit, transmissive, transflective, etc.) simply by swapping out layers. On some screens you can even just flip one of the layers around to switch from one mode to another, although the results are likely to be far from ideal.