Creating curves in after effects

Each layer you convert 3D will become a plane that you can arrange in 3D space. If you convert an vector layer to a shape layer you will get editable vertices  but they will all be in the same plane (X, Y). There is no Z value for a vertex (point on a path).

To edit the paths just select the pen tool (g) and then use the Alt/Option and Ctrl/Cmnd keys as modifiers to select and edit the points on the path. Maybe this video will help a bit:

I apparently misunderstood what you were trying to do.There is nothing in the example illustration that can't be recreated on a standard 2D layer. What exactly are you trying to accomplish.

I would create that type of 2D artwork in Illustrator. There are many more design tools available there and properly layered you can easily create amazing animations directly from AI files.

foughtthelaw suggested converting the AI layers to shape layers. That is only a smart choice if you intend to use shape animators on the layer. Converting an AI layer to a shape looses many of the stroke and fill options, blends, and a host of other design features not available to shape layers. The only advantage in converting to a shape layer is the use of things like Trim Paths, Repeaters and the other shape layer tools. Depending on your design it may be necessary but in my experience 99% of my Illustrator art remains unchanged. Only on rare occasion is it necessary to convert a AI vector path to a shape layer.

If you need the illustration as 3D artwork then it may require additional planning but it is still fairly easy to do in AE with illustrator art, you just have to be careful how you handle the 3D layers and where you place the camera so the illusion is not spoiled.