When you create a shortcut on the home screen to a bookmark, Android will use a apple-touch-icon-precomposed if present (but not apple-touch-icon) as a high-res icon:
<link rel="apple-touch-icon-precomposed" href="https://stackoverflow.com/custom_icon.png"/>
As for the rest of the features, I don’t think there’s any way to do this at present without publishing an Android app that acts as a wrapper for your website.