I would especially like to synchronize Firefox OS with my Lightning calendar. As far as I know, you have to publish your Lightning calender online to make it accessible for Firefox OS.
You can publish your Lightning calendar only on your home intranet and then sync with the calendar app on your Firefox OS phone over your home wifi. Once you step outside your home, the calendar app on your Firefox OS phone will continue to show you your calendar entries because it has offline support.
https://github.com/mozilla-b2g/gaia/tree/master/apps/email
https://github.com/mozilla-b2g/gaia-email-libs-and-more
What exactly do you mean by syncing Thunderbird with Firefox OS apps?