Your idea with the curve isn't bad at all, but for the form of the log you need to alter the geometry as well, I would suggest, create the logo as one mesh, create the shape as simple as possible, in this case just use a plane create the outline, cut the angles with the knife-tool and keep it with only quads, it should be very easy to create this mesh ;-)
When you have the finished mesh, create a few shape-keys, where you just move the vertices back to where they should start. Then insert Keyframes to your timeline, to control the different shape-keys. All the best Marco
For a better understanding here is a quick example of how I would do it.
Here is a little Video https://youtu.be/OW1WdF42yrk on how it would work. And here is a dropboxlink to the example file https://www.dropbox.com/s/aas26i1z0t2e5wa/Logo_folded.blend?dl=0
All the best Marco