Music/iTunes syncing with iOS devices is indeed slightly screwed. From my diagnosis, it is because of iOS 13 update and has little to nothing to do with macOS Catalina update.
Connect your iOS device to Mac.
Go to your device syncing app:
iTunes for pre-Catalina; and,
Finder for macOS Catalina.
Select the device you want to fix your music library on....
From what I found there is no way to add an audio file into Podcast application: as of macOS Catalina, it supports only subscribing to RSS feeds. The only way built-in way to listen to the separate podcast episode is to find it via application search and add it separately.
However, I found out that you can listen to any audio files including MP3 the same ...
Using a 3rd part music manager is the only thing that is really going to help. iPod classic support is killed off in recent iTunes on mac and win so you need a 3rd party app to do so, unless you're still running a few versions back iTunes app. I use "transpod" to add and delete music that gets me by.
Something strange is going on... I have a 5th gen iPod, now called classic (no such tag when purchased), 80 GB, software version 1.1.2 and it works fine with iTunes version 18.104.22.168 running on Windows 10 Home 64-bit version 1903 OS build 18362.418, and has never stopped working through all the iTunes and Windows upgrades/changes since iPod was bought more ...
If I remember correctly, iTunes doesn't let you copy media from your iOS device to iTunes; likely to prevent piracy. If you have purchased the audiobooks from iTunes, you should be able to download them back. If not, which it appears you can't, you may have to contact Apple Support.
I found the way (but still you need iTunes to set order on your iOS device).
Right click on a playlist
Select Copy to Play Order to set the default order for this playlist when copied to a device
There doesn’t seem to be a way to customize the Touch Bar for iTunes/Music. Not all apps support customization.
In many apps, like Finder, Mail, and Safari, you can customize the Touch Bar.
Choose View > Customize Touch Bar. The customization window appears on your display, allowing you to choose your favorite items
Note how they say “many” ...
FYI, the feature I described was called the 3 column browser in iTunes.
According to other users, it has been removed in transitioning to Apple Music. Best we can do to get it back is submit feedback requesting that, here: https://www.apple.com/feedback/macos.html
As for searching your library, you can recover that by enabling the "Filter" field under View....
While downloading the file through Finder, Open terminal:
%lsof | grep ipsw
This command will show you the open ipsw file (which is being downloaded).
The path to find the downloaded ipsw file is:
/Users/Username/Library/iTunes/iPhone Software Updates/