How to run ASP.Net Web Site outside of visual studio 2010?? no exe file?

Question 1

You seem to be fundamentally misunderstanding what a web application is. It's not a standalone executable, that's a desktop application. A web application, by definition, requires a web server. Most commonly for ASP.NET applications, IIS.

Visual Studio provides a built-in web server to use for debugging. Outside of Visual Studio, you deploy the web application to an IIS instance and can then access the application by pointing your browser to that IIS instance.

It can be on the same physical machine as the browser, there's no restriction that a "web server" needs to be a physically separate "server" machine. But you do need a "web server" running somewhere to deploy the application to it.

To sum up:

I now need to be able to run my website on the browser

The browser doesn't "run" the application. The browser connects to the web server, which hosts the application.

I dont want to use the IIS approach is there and alternative?

There are other web servers you can use besides IIS (such as Apache with mod_mono perhaps), though I imagine IIS is going to be the simplest and most straightforward in this case.

Why is there no exe file in my bin folder?

Because you created a web application, not a desktop application.

Can I create one?

No. At least not in the way you think.

Also whats with the dll files?

Those are where compiled code is kept. A web server would use them to invoke functionality in your application. Indeed, a desktop application would also use them to invoke that same functionality.

Question 2

I dont want to use the IIS approach is there and alternative?

Yes, you can use Owin and Katan. It creates an executable file.

By and large, you run the exe file on server. Then client browser accesses the application.

Note: If you are new to ASP.net, it is not what you want.