Run node.js app with node-webkit?

Yes, it is. Node-WebKit is a package with browser + node web server. But I believe you will probably want to change your architecture, because you don't need the client-server style. If you run your app without any changes, node-WebKit will act just like node server and you still need a browser to access the app. There is no reason to use node-WebKit instead of pure node in this case.

To use node-WebKit embedded browser you should know that there are no need to start node server. The browser's JavaScript environment is already connected to node and you can execute node commands and packages direct from the JavaScript files (eg.: access file system from the browser, a dream to every web developer). It's like you are running a browser inside the server, without the need of make requests and receive responses... For this reason you don't need to use packages as socket.io, cause the communication is already established. But you can use the fact of node is a server to easily establish communication between different machines, for example.