I have a ASP.NET MVC6 web application developed in VS 2015. I run it on Windows 10. I have client side JavaScript compiled by Gulp which is dropped in the wwwroot folder which I want to debug as well as debug the WebAPI controllers on the server side. The application will be hosted on Azure after the development is complete.
During the development, should I use (1) WebDev.WebServer.exe (2) IISExpress.exe (3) IIS10 or (4) DNU to host the ASP.NET website?
IIS Express (2) is good enough for about 95% of things you need to do during development, it's as good an implementation of "proper" IIS7+ as you'll likely need.
That said, for things you can't properly emulate on your workstation, such as Azure, periodic deployment to your Azure environment is always a good idea, just to be sure you don't fall into the trap of "Well it worked on my machine".
Never use the built-in webserver.exe, it's a toy and is severely limited in functionality.