This is already supported and has been since VS2005. It's not that well-known though. There are 2 web project systems in VS - one is a project file (.csproj) system and the other is a folder based system.
In Visual Studio, go to File -> New Website... to create a folder based web project. It will automatically do the correct nesting if you have Web Essentials installed (it's a free download).
1) This doesn't effect anything in your app at all. Either runtime or design time. All it does is to change the visual representation in solution explorer. It does write to your .csproj file to make it work, but it doesn't have any functional impact.
2) In the .csproj file
3) The rules are not saved in the project. You can find them in Tools -> Options -> File Nesting
I'm not sure. I don't think I will since this is not a web specific feature. However, I said the same thing with the Image Optimizer extension and it's now part of Web Essentials.
Make sure the rules are enabled by going to Tools -> Options -> File Nesting and setting them all to true.
That is my plan but so far I get errors when I try. I haven't given up though, but if you know some tricks, please see if you can make it work. I've already made it so the project can be opened in VS2012 (you need the VS2012 SDK too).
The .vtt syntax highlighting is coming to both Web Essentials 2012 and 2013 this weekend. The CSS support for ::cue is already out, so Web Essentials should have automatically downloaded the updates to the CSS Intellisense and validator.
VS2013 Preview doesn't have the Browser Link extensibility API, so the Chrome extension wouldn't work with that. All my demo extensions will be open sourced as soon as the final release of VS2013 is out.