Installing system software on my iMac desktop computer

First make sure you have a copy of all your changed data before you reinstall the older macOS since you will need to erase the drive in the process.

You can either reinstall an older version of macOS or you can restore from a backup you made before the Catalina upgrade. Both methods require you to erase the whole physical drive. If you have a pre-Catalina bootable backup, then boot from it, otherwise you will need to reinstall macOS.

See if you can download Mojave and create a USB installer:

https://support.apple.com/en-us/HT201372

Personally I have not been able to download any previous macOS installer from Catalina, so if this is the case for you, then you can try booting into Internet Recovery Mode using Command + Option + Shift + R to load the original OS which shipped with your Mac. Once you install the original OS, you can download the Mojave installer.

You will need to erase the whole physical drive before reinstalling macOS. If the physical drive is not shown in Disk Utility, then you will need to click on "View" within Disk Utility and select "Show All Devices" so that the physical drive appears in the left pane of Disk Utility. Erase the drive as GUID partition and MacOS Extended (Journaled).

I don't use Time Machine, but it appears you must boot from a macOS installer or from Recovery Mode/Internet Recovery Mode to restore from your TM backup using these instructions.

https://support.apple.com/en-us/HT203981#macos

Just make sure to have copies of any new documents or changes since the pre-Catalina backup or you will lose those changes or new documents. However, after booting to your TM backup you will need to launch Disk Utility and erase the whole physical drive as I described in my previous post. If you do not erase the whole physical drive, then you will have remnants of Catalina hanging around in the form of multiple volumes.

If you are restoring to High Sierra or Mojave, then you need to erase the drive as GUID partition and APFS (top option).

For Sierra or earlier you need to erase the drive as GUID partition and MacOS Extended (Journaled).

Once you have erase the whole physical drive, then you can quit Disk Utility and select "Restore from Time Machine Backup" as described here

https://support.apple.com/en-us/HT203981#macos

As for the Mojave download, I'm not sure what is happening since I've been unable to even initiate the download from Catalina. As I mentioned if you cannot successfully download Mojave and create a USB installer, then you can use Internet Recovery Mode to boot to the installer for the original OS which shipped with your computer. Once you have the original OS installed you should have an easier time downloading the Mojave installer.

I'm glad it all worked out for you.