В Microsoft об этом тоже думали, наиболее вменяемый на данный момент концепт, как ни странно:
research.microsoft.com/en-us/projects/touchstudio/

Нельзя.

developer.android.com/reference/android/webkit/WebView.html, Cookie and window management:

For obvious security reasons, your application has its own cache, cookie store etc.—it does not share the Browser application's data. Cookies are managed on a separate thread, so operations like index building don't block the UI thread. Follow the instructions in CookieSyncManager if you want to use cookies in your application.