Hello Oystein,
Good questions. We do have a Facebook Events add-on that will import events from Facebook to The Events Calendar–just not the other way around. There are other plugins out in the wild that will send events from The Events Calendar to Facebook. Here’s one, for example.
Posting to Google Calendar is a little different. The Events Calendar does include a URL to the calendar’s ICS file, which can then be entered into a Google Calendar to push events there. However, this is not a continuous sync and would only make the push once for each time the URL is added.
Does this help answer your questions? Please let me know. 🙂
Cheers!
Geoff