I am a developer that creates apps for Windows Phone. I also create an app "Week View" that's provides a weekly view of appointments as the name suggests.
Please let me explain, how calendar apps for Windows Phone work.
Do "all" calendar apps on a phone share a common underlying calendar datastore provided by the phone?
No. The "calendar store" is in the cloud. The native calendar for Windows Phone is "Windows Live". You can also log in to Windows Live in your browser and create appointments in it. When you setup your phone with the Windows Live calendar, it automatically syncs with it.
So the phone shows and keeps the synced data but the storage for all of your events is the Windows Live calendar is "in the cloud". By the way, this is nothing special - it's the same thing with Google Calendar/Android and as far as I know iOS/iCloud calendar.
Major advantage: Access your calendar on your browser and everywhere (where you have an internet connection) and your appointments are not lost when you lose your phone.
How do they sync?
Windows Phone automatically syncs periodically with all sync enabled calendars. This is done by the OS, not by an app.
Should a calendar app be able to do the same things (write to, creating and modifying stuff in the calendar) as the built-in calendar?
The bad news is: MS has not fully opened the API so functions that the built in calendar can do are not available for developers. In fact the API has many limitations. Editing/deleting appointments is not possible (creating appointments IS possible in WP8), no details for private appointments are returned, no calendar categories, ...
To bypass these limitations some of the current calendar apps (including "Week View," of course) use another way: Accessing a users calendar by using the Windows Live SDK. This allows the same options as accessing your calendar in your web browser.
This way calendar apps, such as Week View, can create/edit/delete appointments in the Windows Live calendar, show details for private appointments and calendar categories.
Week View has a free trial if you would like to try it out.