Creating a Star Map
I'd like to create a planar (rectangle shaped) map of the entire (both hemispheres) sky, with stretching anything (keeping all constellations etc. as seen in the sky).
This is a physical impossibility. You simply cannot map a spherical entity (the celestial sphere) onto a plane without introducing some distortion. Cartographers have developed many different projections in their efforts to solve this problem, but none is perfect. All of them are forced to introduce distortion at some point.
The Mercator projection suggested in another answer is notoriously inaccurate as you get closer to the poles, making Greenland as large as South America, and stretching Antarctica into an impossible shape.
The best solution is to forget about mapping the whole sky, but to concentrate on smaller areas where the distortions are not as severe. This is what most star atlases and planetarium software programs do.
As Geoff said, what you ask for is a mathematical impossibility. That being said, however, there are LOTS of ways to draw a map, and some come closer to what you are looking for than others. This article provides lots of general background on map projections. You will want to pay particular attention to the section that discusses classifications.
It sounds like you would be most interested in the Interrupted Goode homolosine projection, applied to the sky. Obviously, you wouldn't want the same interruptions for the Earth and the sky, but that's probably your best bet.
what you're looking for is a Mercator projection is a cylindrical map projection type of flat land to develop but can be applied to the stars.
Mercator, by projection, intended to represent the Earth's spherical surface on a cylindrical surface tangent to Ecuador, which generates a map to deploy terrestrial plane.
It is an idealized model that treats the earth as an inflatable balloon that is inserted into a cylinder and begins to "inflate" the cylinder volume occupied by printing a map on its outer face.
This cylinder cut lengthwise and deployed would be the representation you need.
this are the ecuations that you need to do representation:
if $ \phi $ is $AR$ and $\lambda$ is $DEC$, then:
$$ x = \lambda -\lambda_0$$ (being the length $λ_0$ center of the map)
$$y = ln [tag (\frac{\pi}{4}-\frac{\phi}{2})] $$
