"actual order distance"
Do you mean the distance between one order to the next? If yes, it is available in the Orders attribute table of the VRP analysis group layer.
Distance traveled for only on road segments that have orders.
Not counting the road segment again if need to traverse it again as part the route to get to next order.
Take a look at the GP tool Copy traversed Source Features.
You run this after you solve the VRP and it will write out all the edges that were traversed by all the routes. You can select only the ones that have orders on them. You can do this either by spatial select or perhaps using the FromPosition and ToPosition are not 0 or 1 (unless the order snapped to an end point). Once you have that edge, you can get the distance from a field like Attr_Meters.
I have generated edge and used that tool.
I will look into your suggestions. Spatial select will not work for me since it may select the to and from route road segment. Essentially doubling the distance. I will try the 0 and 1 approach. Fortunately, I don't have any orders on an endpoint.