Our product is an ASP.Net web application. Currently, we use Web Site Projects in Visual Studio, but have been looking into using Web Application Projects for quite some time.
If i have correctly understood your question, you'd like to publish also items which aren't compiled (htm, JS, images, etc..).
So, each file in your Solution Explorer tab has its own properties (accessed by F4 key) which let you choose the build action (eg. compile -> will inject the item in the DLL if applicable, content -> will copy the file "as is" to output directory).
I think the build action "content", with the option "copy to output directory" set to "copy if newer", may be the solution you're looking for.