I also see the same symptoms. FP3, latest /e/ 0.16.
In my case, authentication for the mail app fails occasionally. Sometimes I can receive and send mails, sometimes not.
For the calendar, things look more predictable. I do have several calendars in my google account. The app see all of them, but cannot sync the primary one due to failed authentication. All other calendars seem to be OK.
My google account is protected with 2FA. I used the “Login in other session” method to authorize the /e/ apps.
Is there any chance to debug this?
Cheers,
Michael