This is a bit confusing.
Probably you did the connections right but writing it down here became salad ?
1) I connected the ADAT out (main) of the ADI-8 to the ADAT in of the UC and the ADAT in (main) to the ADAT in.
It should be this way with two toslink lightpipe cables :
First Toslink cable connected From ADI8-input to the UC Adat output.
Second Toslink cable connect from the ADI8-output to the UC Adat Input.

Open the program Fireface USB settings,
clock mode : set to external adat.

Under your point 4) Source: AES change this to Adat

Ok, stop and thinking for a moment.
What do you want to do exactly ?

What is your desire for having this ADI8-DD ???

3) XLR male to TRS balanced connectors (four of them) from the  inputs (ADI) to a patch bay. XLR female to TRS balanced connectors (four) from the outputs (ADI) to a patch bay.

What do you want to connect to this patchbay ?
XLR to TRS balanced cables ?
Hä !?
OK I think you are complete  lost on the wrong Autobahn you Geisterfahrer !

Your device ADI8-DD is a digital format converter and not an analog converter !
You can convert the UCs adat digital 8-channel format into AES/EBU.
This is useful for connecting digital devices like multieffectprocessors Eventide H8000 or Eventide Eclipse for example.
Those devices have AES connection (XLR connectors) and it is cool to have them in the system digitally only, without analog conversions.
Speaking of TRS balanced cables, this means you want to connect analog gear, may be some synths, keyboards etc.
If so, bring back the ADI8-DD to the sales store and try to get an ADI8-QS or an ADI8-DS or an ADA8200.
Those are analog converters with 8 Ins and 8 Outs for TRS cables.
Connection with the UC is the same.

Adat Out goes to Adat In of the other unit and wise versa.
The converter is the master ( I learned it that way) clock internal.
All other devices are slave, clock external adat input.

Short and compact Newtons Law of Adat !

Behringer ADA8200. Less than 200$. Midas preamps. 8 in and 8 out on XLR.

The Focusrite OctoPre is analog inputs only. The analog outputs carry the signal from the mic preamps. Great if that's what you need, but lots of people think they're from the ADAT input.

Several: If you are investing, several RMEs, ART, Presonus, Ferrofish, Audient, MOTU, Arturia...

I also noted the newest Focusrite Clarett OctoPre has 8 analog outputs from ADAT:

https://www.thomann.de/be/focusrite_clarett_octopre.htm

Sorry for the confusion. Previous ones didn't have those.

RME:
https://www.thomann.de/be/rme_octamic_ii.htm
https://www.thomann.de/be/rme_adi8_ds.htm
https://rme-audio.de/adi-8-qs.html

The QuadMic is just a preamp. No AD.

https://rme-audio.de/quadmic-ii.html

My budget doesn't allow the higher end RME's either. That's why I suggested the Behringer.

I currently have 2 ADA8000 units. The predecessor of the ADA8200. They're fine, as long as you don't stack them in a rack. They do tend to get fairly hot.

I also have four Fostex VC-8s. But I wouldn't suggest these, as they are 16 bit only and have no mic preamps. Also, they're consumer-level in and out, on RCA plugs.

These are used with 2 appsys.ch boxes that pass 32 channels of audio over utp wiring. And a mixer that has 16 channels in and 16 out over ADAT. And 2 Macs, one with a FF400, the other with an old M-Audio FireBridge.

All that can be switched over easily, per block of 8 ADAT channels. At a distance of up to 100 meters, if needed.

I'm a bit of an ADAT nut I'm afraid.

Behringer ADA8200. Less than 200$. Midas preamps. 8 in and 8 out on XLR.
The Focusrite OctoPre is analog inputs only. The analog outputs carry the signal from the mic preamps. Great if that's what you need, but lots of people think they're from the ADAT input.

Hi Ingo
This is very important to understand !
The device has probably 8 XLR sockets on the back ( Outputs ) but they do not deliver sound from the computer.
There must be an Adat Input socket for getting computer sound.

indeed there is not a big number to choose from.
Ada8200
Clarette Octopre
ADI-8-DS
ADI-8-QS (QS ? Quantum Supertechnology from Aliens)
Ferrofish Pulse 16

My tip ADI-8-DS if you can find a used cheap one.
And the ADA8200 ( I tested myself) is surprisingly good !
Don't be afraid of the brand name Behringer, they changed a lot in the past 15 years.
It is a Swiss/German company with own factory in China (3000 employes ! )
Get an RME unit and you will learn a lot about the analog to digital music technology and it is a big fun to receive great sounding signals.

> One more general question: Some ADAT interfaces such as the Focusrite come with their own software.

ASIO driver / Mixcontrol software are only available for "recording interfaces" where routing and
the connection to the PC takes place.

> Will this software interfere with Total Mix?

Not applicable in this case. No you can use any other driver additionaly to control the routing of another recording interface.
BUT .. a DAW can only load ONE ASIO driver .... and this will be the (better) one from RME and where you have
a much better Mixing Software (TM FX) compared to Focusrite Mixcontrol.

> Or do I just leave any e.g. Focusrite software uninstalled? Or do they live in peaceful coexistence?

For a preamp there is no mixcontrol software and no ASIO driver.
Look at the back of this device .. there is not even an USB port.
All the communication in and out happens through ADAT.

As I wrote earlier: I am from the past. ;-)

However, just to make sure that I got both of you right: I could connect e.g. the Focusrite via ADAT. I do not even have to care to install the Focusrite software. The Focusrite will be part of my set up via ADAT (physical connection) and TotalMix (software based routing)? That would be my prevered way to go anyway.

I am a sorry for being such a noob. However, I have learnt quite a lot from your answers.