Thunderbird has native CalDAV support, no add-on is required, but it doesn't setup the calendar automatically for all providers. You will have to manually add your online (remote) calendar to Thunderbird. The default calendar in Thunderbird is a local (offline) one. You can check if a calendar is a local or remote one, by right-clicking the calendar and selecting Properties. An online one will have options like Location (its url), username, refresh interval etc, that only apply to online calendars. An offline (local) calendar will not have such options because it doesn't reside on a remote server. It only exists locally, on your computer.

Thank you very much for this help. Now I understand much more how the system works. My Thunderbird calendar is now online, but I still cannot see the content on my iPhone even though it's connected to the CalDAV account. In Thunderbird, in the calendar <right click> properties window, there is a box called 'Location'. It says now 'https://caldav.domeneshop.no/calendars/..... '. This address might be wrong. Do you know how I can change it?

You can't edit the Location field after the calendar has been added. Remove the calendar (right-click > Unsubscribe Calendar) and re-add it using the correct address.