Hi @nikhil jain ,
1) The document says an ASE gets created automatically once a App service is created within an App service plan. But it didn't create for me as I was able to build, deploy and test a simple internet-enabled Web application without it as well.
Correct. Creating an ASE does not create an app service. But it will create the app service plan that you can later assign to an app service.
2) Does the costing incur for both ASE and App Service Plan? If that's so, then a standalone VM would be cheaper, i guess for comparable SKU?
ASE costs derives from the app service plan, the isolated plan SKUs us on this page. The two aren't separate
3) Can I create an internal Web app (Not exposed to Internet) without an ASE? if yes, how?
You can add IP access restrictions to your app service. The *.azurewebsites.net will still be accessible but IPs not allowed will receive a 50x when navigated. If you can go one step further and integrate a VNet and have your app service accessible only from Azure Service Endpoints