That said... Things like Phonegap (web views) are a wonderful idea for utility apps because the performance is good enough for those (and hey even some games) and they end up being easily ported across many mobile operating systems.
I would look into Phonegap, Appgyver (a new and totally awesome one because parts of it will utilize the native OS and your app will feel more responsive), Appcelerator Titanium, and perhaps even the new Famo.us one. Also take a look at the Ionic Framework for some further ideas about mobile UI and what you can do with these web view style apps.
I'm a developer at Clarity and we use PhoneGap to publish our native iOS application to the App Store.
From experience, wrapping our website in PhoneGap has been a really good way to get a native application that has native features like push notifications for "cheap".
The nature of Clarity itself requires you to be online to use the product: browse and search, request calls, ask questions, etc. For that reason, we never looked at providing offline support. If we can't find an internet connection, we show a very simple alert dialog.
Most definitely! There is a WordPress plugin called AppPresser which connects WordPress to PhoneGap, making it easy to access native device features and create an application that feels less like a wrapper. One of the challenges mentioned above is offline access, however, that is currently being worked on. Currently, there are a number of extensions to enable native device features as well as adding e-commerce functionality via Woocommerce.
AppPresser requires an app-specific theme to work (you can always build your own) but fortunately, there is the AppTheme provided by the makers of AppPresser, which offers an optimized app-style theme that is more likely to be approved by Apple's review team.
You can see AppPresser in action (there are a few apps available now using it) by checking out their website: http://apppresser.com.
No you can't encapsulate it.
You can crept APIs for your content or implement a theme that is responsive and looks good on mobile web but there is no way of directly wrapping your existing code.
Also, a part from the technical side, Apple always tries to reject apps that are purely copies of websites.
Please make good apps, that do something.
We've actually accomplished this with the Marketing Technology Blog through a third-party platform that optimizes the visual content and lays out the posts and pages in an application that's beautiful but never requires updating. We did this with a company called Postano who have several clients on it. We're working on actually partnering with them to deploy it to more people. The platform they built connects to your WordPress site but outputs native applications for both Android and iOS (and others). It's not using PHP OR WordPress - just driving and optimizing the content FROM WordPress to a third-party platform for content optimization.
If you wantto make a mobile app stay far away from WordPress. Many of the plugins and tools will become an issue when WordPress does an update an kills everything.
Phonegap is OK but not optimal. I suggest you create a web app using the M.E.A.N. stack. If you don't have the tech talent or budget, you can submit your project to an organization like www.AlchemyLab.co and their students can develop it as part of their projects.