Can't install Synaptic Package Manager

Try searching for just 'Synaptic'.

Also, on the Software & Updates > Other Software tab, select each line that has a check-mark, then select  Edit , going through each check-marked entry to ensure yours looks like the entries that I put in my last post.

The entry:

Type: Binary
URI: http://ppa.launchpad.net/kelebek333/mint-tools/ubuntu
Distribution: focal
Components: main

... is related to Warpinator, which I use to transfer files to/from my Android phone. So you can probably ignore that one.