I can answer the second part of your question. Since you just updated the MacBook's iTunes settings to point to the NAS, it's iTunes library file still resides on the MacBook. This is a .itl file in your ~/Music/iTunes folder.
iTunes relies on this database to tell it what's in the media folder. If you add stuff to the media folder without adding it through iTunes (i.e. just dropping files in Finder, or using another copy of iTunes on another machine) then that .itl database does not get updated.
So when you add files through iTunes on your Mac Pro, it updates the .itl database on your Mac Pro, but not your MacBook, so the MB can't see those files. If you add anything through iTunes on the MB, the Mac Pro database won't have it, since the copy of iTunes that connects to that database did not process the files.
I have a similar setup to what you're doing: iTunes media on an external drive connected to my iMac, and a MacBook which I use ONLY Home Sharing on.
You might get the idea to move one .itl file to the NAS and open it in iTunes on both machines, and this will work, but you can/should never have it open in both at the same time. That .itl file is really just an SQLite database, and they don't allow simultaneous access, so I've never even attempted this.
With respect to your first question, check ~/Music/iTunes and see if there are any old library (.itl) files. It's most likely that iTunes is opening an old copy or a backup of the database. If you duplicated the database or started a new one before moving your media, this might be the cause.
Start iTunes and make sure the media folder is set correctly or make some other change in iTunes. Now go to ~/Music/iTunes and look for the .itl file that has most recently been modified. Move all the others out of the folder.
Quit iTunes and restart it while holding the Option/Alt key. This will bring up a dialog that lets you select which database to use. Select the .itl file you left alone and it will continue using that as the default library.
Best Answer
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 way you did in Podcasts in Books application by adding them there via
File->Add to Library...and then syncing them to iPhone.