How can automatically I stop the development web server when I stop debugging in Visual Studio?
There is no way to get the Asp.Net development service to automatically stop when you hit the "stop debugging" button, as all that does is detach the debugger. You could however explicitly kill the process yourself (which would in turn cause VS to stop debugging).
This question shows how you might do this in a Macro:
Automatically stop/restart ASP.NET Development Server on Build
Alternatively take a look at mat3's answer on how you can force the development server to restart every time you start debugging (although it will still remain running after you have finished your debugging session until you next run your probject).