Looks like your culvert may be a little undersized. You should probably install some sort of brush guard on the intake side of the culvert, to prevent blockage in the event of a heavy rain. I'd cut those trees down on the right, as seen in picture #2, and make the earthen bridge as wide and high as possible to compensate for future erosion and subsidence. And be prepared to occasionally add more topping material as needed.
If this is to be a fulltime driveway for a residence, you were well advised to use something more substantial than topsoil as a base.
