What OS did you manage to get reinstalled on your mac.
When you restored from your back up what files did you restore
you should only restore your personal data, restoring System and Library files
would just put Catalina back on, which you do not want.
Erase your mac again.
Boot to the Recovery HD, open Disk Utility.
Click View in the menubar (if it is available) select Show All Devices.
Highlight the Disk not any of the indented Volumes.
Click Erase
Give the Disk a name
Format: Mac OS Extended (Journaled)
Scheme: GUID Partition Map
Click Erase.
When Done quit Disk Utility.
Now reboot while pressing Shift Option Command R
to reinstall the original OS.
If you get the original OS installed do not restore anything from a back up yet.
Make sure the mac is running well with a basic install.
Then you can think about upgrading it to Mojave.
Install Mojave and run it to see if it is working well.
Then when you are happy your mac is okay then you can decide to restore from the back up.
Remember only restore your personal data, do not restore any System or Library folders or files
that will only put Catalina system and library items back on your mac, which will cause trouble again.
Do not restore any apps, these will put the Catalina version of apps back on your mac, they will
not work with an older OS.
Think and plan your strategy, don't rush at things.
In future if you experience problems with your mac, it is better to sort them out.
Erasing and reinstalling is not the best plan, look at the mess you are in,
and the Apple Recovery Servers are unreliable.
If you are going to erase and reinstall make sure you have a bootable
USB installer with the OS you want to reinstall to hand do not
rely on outside agencies.
When you are up and running with the older OS you will need to
download Install macOS Mojave.app, get it here.
Open your Safari browser and click below.
How to get old versions of macOS – Apple Support
Go to Download OS click on macOS Mojave
This will direct you to the AppStore Mojave download page
it should say Get or Download.
This will download Install macOS Mojave.app to your Applications folder
it should be 6.05GBs in size.
Now you can use that to make bootable USB installer using the instructions here,
How to create a bootable installer for macOS – Apple Support