The canonical fix for troubleshooting Mail on OS X is to run the Connection Doctor (it's slightly buried in the Window menu without a keyboard shortcut to summon it) to isolate whether you have a general network error condition or a specific error on one or more of the accounts.

Each mail account will typically have two servers - different internet ports and potentially different username / password / authentication details.

Once you can isolate a specific error message on a specific port to a specific IP address, you can begin to troubleshoot whether you have a configuration issue, a network reachability issue or a server issue.

The show detail button is helpful to log the exact transactions and communications between Mail and the server.

If you have a vexing problem, I have found that disconnecting the network connection and quitting Mail (or forcing a reboot while the network is off) will let you start Mail and disable all Mail accounts except for one and also check for a stuck message or two in the Outbox. File those away as drafts so that you can then run connection doctor, re-enable the network and systematically work on the communication needed between the Mail program and your mail server.

Possible reason for the problem

I am not 100% sure but my guess on why all this happened is the special way I set up my disk encryption. (For detail: I used normal File Fault encryption but because my standard user has a weak password I removed this user from the users who are able to decrypt and instead added a root user with a strong password who was able to decrypt the drive.)

As my research provided the the update to High Sierra changes the file system format from HPFS+ to the new standard called APFS. With a different encryption as well. My guess is that during the upgrade either the special root user was removed or another error occurred in connection to this constellation.

How I finally dealt with it

As written above I was able to boot in the recovery mode of the TimeMachine disk. But I had to realise that this time machine was running OSX El Capitan (obviously the running system of my Mac when I created the first backup) and El Capitan was of course not able to either decrypt or even identify the new APFS file system on my internal drive (which explains the missing partition in my picture).

The next try was (don't know why I didn't try this earlier) to boot in normal recovery mode by holding down CMD+R while booting. This worked and I was able to use the disk utility to delete the Macintosh HD partition, create a new one instead (as APFS) and the reinstall macOS High Sierra from the recovery menu.

After the complete new installation finished I was asked if I want to move any data and I selected my Time Machine backup. It needed a few hours but after this my system is back on track. Everything seems and where I left it.

Notes

• As general note for all of you who want to perform the upgrade: Back up your data immediately before doing it!

• And after the update is done shut down the Mac once to check if everything is fine. (Don't wanna think about the scenario where I would have worked a few days without shutdown any new backup and then shut it down to end with this...)

• After my described fix I had to set up the disk encryption again (now I only use my normal user and instead of any trick the account now has a strong password...)

Thanks for your help :)