Yes, all emails are forwarded to the Continuity Service.

What conditions must be met for all emails to be forwarded?

The Continuity Service must already be activated before your destination server fails. Please note that the initial activation can take up to four hours, as during this time the mailboxes and the associated databases are created. Only then can you log into the webmail in the event of a mail server outage.

Email traffic looks like this after successful activation:

In the event that your mail server is unreachable, emails are automatically delivered to the webmail. You can retrieve and view them via the webmail. When your destination server becomes reachable again, the emails are deleted from the webmail and automatically delivered to your destination server.
Why emails continue to appear as delayed in Email Live Tracking after delivery can be found here: Emails in the Continuity Service continue to show as delayed after delivery

If you want to know why an email is not forwarded from the Continuity Service to your destination server, please read the following article: Which emails can be delivered by the Continuity Service?