How to display text on a cube and how to change it dynamically?

Hah, changing the cube texture IS what I'm after, probably.. and there's no need to fix the unwrap, it's fine.

I've been thinking about smthg like this:

 Texture2D texture = null;
                     Color fillColor = new Color (1, 0.0f, 0.0f);
                     Color[] fillColorArray = texture.GetPixels ();
                     
                     for (int m=0; i<fillColorArray.Length; ++m) 
                     {
                         fillColorArray[i] = fillColor;
                     }
                     
                     texture.SetPixels (fillColorArray);
                     texture.Apply ();
                     material.mainTexture = texture;
 
                     GameObject.Find("object_name").transform.renderer.material = material;
 
                 

but how to put a text into that texture? And how to do all this from a code?

Well, like I said, I think putting it directly on the texture would be a huge pain. There's no easy way to access a letter texture, much less a whole word; or put it on only 1 side; or scale/rotate/position the word pixels onto the mainTexture pixels.

If you have a few fixed phrases, most people would just pre-make textures with those, get them all nice, and swap textures (or even materials) in-game.

Older material, but it should still work:

http://blog.almostlogical.com/2010/08/20/adding-text-to-texture-at-runtime-in-unity3d-without-using-render-texture/