Culverts in HEC are modeled using FHWA standard equations for culvert hydraulics. They can be partially filled with material, which give the culvert a flat bottom.
In my state, we have a stream crossing guideline that encourages/requires stream simulation culverts. There has to be a channel in the culvert matching the upstream and downstream channel. For this reason I use a bridge with abutments to model a box culvert. I believe using a culvert routine would misrepresent the channel cross section within the culvert.
A storage area with pumping could be used to model diversion and dewatering. A partially obstructed, upstream culvert may serve as a coffer dam. Be careful not to fully dewater the channel if the model is unsteady.