ArcGIS Pro is packaged with the ArcGIS Desktop suite. When you buy the Network Analyst Extension for Desktop, you get the same extension for Pro. However, the license itself is separate. There is a NA extension license for Desktop and an NA extension license for Pro. Basically you will be provided a separate authorization number for each application.
As for ArcGIS Server, that is a separate product and has its own set of licenses. So yes, you will need to purchase an ArcGIS Server extension license. However, I don't think you need to purchase a license for all three environment. You only need to purchase the license for the Production environment. Since your Dev and Staging are not public, you should be OK. I suggest you contact your account representative for verification.
