With AngularJs + Web API, why we need ASP.NET MVC and IIS?
Can I just use AngularJS plus WEB API to develop a web application without IIS and ASP.NET MVC ?
It is definitely possible from the technology point view. I see 2 areas of concern:
Would clients with outdated browsers have problems accessing the site? This is less of a problem as new versions of browsers are getting released.
Are the non-IIS hosted hosted Web API services robust and scalable enough?
With AngularJs + Web API, why we need ASP.NET MVC and IIS?
By off-loading all the logic onto the client without providing a server side fallback, you cause the site to break when the JS doesn't load, including for search engine indexing bots.
Can I just use AngularJS plus WEB API to develop a web application without IIS and ASP.NET MVC ?
Yes, it will just be fragile and invisible to search engines.