After playing with the thing for hours, I realized that Eric's solution is not working as intended because when Request.Url.DnsSafeHost (C#) is accessed, it gives me localhost instead of the used domain name, which is not acceptable for my application development. I found the solution with Eric's hint though:
Full step by step for using custom domain for local ASP.NET development machine with IIS Express:
1. Install Telerik Fiddler, run and edit Tools -> HOSTS and assign "localhost:[port number] domain.name"
2. Open [User Folder]\Documents\IISExpress\config\applicationhost.config and find the application <site> setting entry.
3. Add <binding protocol="http" bindingInformation="*:[port number]:domain.name" /> to the <bindings>
4. Run Visual Studio as Administrator, build and run the application.
Thanks again Eric :D