When I morph the A curve into the B curve (target), and set the Shape Key value to 0.5, I get the C curve. If I duplicat the C curve, I get a bezier curve of a A' / B' curve, not a simple bezier curve. Is it possible to get the simple bezier curve?
If you mean you want the "C" state to be the original state of the curve without the use of shape keys, then just set the shape key to the desired Value (like 0.5 in your example), then click on the Shape Key Specials menu button (the downward arrow) and choose Apply All Shape Keys.
There are 2 ways to duplicate an object in Blender. Shift + D duplicates an object making a copy of the object itself and a copy of the original object's data and Alt + D (duplicate linked) duplicates only the object, but links the original object's data to it so that when you edit that data(for example the geometry), both objects sharing that data change. So if you want the duplicated curve to be separate and unique, you should use Shift + D for simple duplicate(as opposed to duplicate linked).
There are a few ways to "morph" a curve, but if you use shape keys to do it, if you duplicate some object that has shape keys, they will also be duplicated with the object. You can however apply shape keys to any object that has them(duplicated or not) by choosing Apply All Shape Keys From the menu in the Shape Keys panel, Properties Editor, Data tab:
Note that if you have a bezier curve and use shape keys for it, it is still considered to be a simple bezier curve. There is nothing called "morph" curves or objects in Blender.