How to call (or use) saved project (.vcmx file) in .NET and how to run the simulation (Once the VC opens) using .NET API code?
Take a look at ISimulation and the ISimulationService. As for loading saved project, look at IApplication and ISimWorld.
Addition to zesty’s hints: You can create a plugin that runs the simulation automatically once a layout is opened. And if you have associated .vcmx files to open with VC it should open that layout automatically (by double-clicking in file explorer) once VC has initialized.
So all you need to do in the plugin is to use LayoutLoaded event of IApplication and to start the sim, RunSimulation method of ISimulationService.
For creating such plugin refer to: