Do I need ArcGIS Pro if I want to design my layout templates in VSP Designer and use them in VSW applications?

You do NOT need ArcGIS Pro for designing layout templates in VSP Designer and using them in VSW applications. If you use the VSP Designer in ArcGIS Pro Layout Mode you can create, update, open and import ArcGIS Pro layout items in your browser. ArcGIS Pro layout items can also be created by using the ArcGIS Server environment. ArcGIS Pro layout items can be referenced in the print requests of VSW applications against the VSP Engine Tools, which are generating the actual print. ArcGIS Pro is only recommended for installing the VSP Engine Tools.