I figure what is important now is
Can you access the TC and recover your files.. ?????
Once you do that you can ignore the whole lot and start again any way you desire..
I need to know everything to help.. so far you have not actually given us sufficient network info to help with network.
. Do you still think Migration Assistant will work? I haven't tried it yet.
Do or Do NOT .. there is no try.. master yoda.
In the world of pain called Time Machine there is lots of trial and pain. So throw out all your Force indoctrination... it is useless not to try.. so try.. and try and try some more.
Chose Extend as the option for the Airport Extreme, which created the extra password.
You are introducing new items into the discussion all the time. Give me everything.
I need to know what model your main router is and how it is configured.. Screenshots of what you are seeing in airport utility helps me enormously.
What model is the Extreme and how it is configured??
What model is the TC and how it is configured??
Screenshots from airport utility assuming it now works helps no end.
It is perhaps time to move the TC directly to plug into the Mac Mini and see if you can access it for recovery of your files. Forget the network outside of that.
Once you accomplish this.. reestablish the network.
For the future .. Forget Time Machine as backup software for Catalina.. download and use Carbon Copy Cloner.. a reliable backup as opposed to Time Machine which is NOT reliable at all.
BTW, the home network and the user with admin privileges on the Catalain MacMini both still have Apple-prompted space characters in their names, so I should change those two as well, then try once more to access TM?
You need to fix the setup in Time Machine by repointing it to the location of the backup.
Just select the backup location now and add it to time machine.. and delete old entries.. they won't work now.
However, I avoided upgrading MacMini from 10.13 due to quirks I read about when 10.14 came out, so maybe just re-install 10.13 if I can find it via App Store?
Yes.. but do it via Recovery as I already explained.. boot to recovery and use set assistant. If the last backup is still available then load it to a USB drive otherwise you are going to totally mess yourself up and us as well..
I reiterate that Catalina is unsuited to 2012 mac mini.. Mojave is likely the best one to stick with. And no more upgrades.