ipod shuffle not showing in iTunes

Make sure the connector on USB docking cable is fully inserted into the shuffle's headphones jack, as shown in this document

https://support.apple.com/en-us/HT203407

The fit can be tighter than expected, especially when brand new.

If that's not the problem, does the tiny light on the shuffle do anything when you connect it? If it does, describe what it does (color, flashing, solid...) immediately after connecting.

Do you know why the iPod is not yet showing as authorized device or does that happen automatically?

I don't understand the question? If the iPod appears in iTunes and you can load songs on it, there is no "authorization" to use it. Are you referring to a message that comes up saying you need to authorize your computer?

however it will only playlists with regular music, not the podcast playlist.

That can be confusing... If you just play the shuffle, it is playing from the main All Songs list. Podcasts are not songs, so they do not play from that list. The current shuffle has a feature called VoiceOver, which you use to select other lists to play from. If you load podcasts on the shuffle, the episodes of each podcast are grouped together and considered a separate playlist. If you load an audiobook, it is also a playlist on the shuffle. And you can also load your own playlists of songs (that you created in iTunes); for example, a list of songs for exercising.

Then, you use VoiceOver on the shuffle to select the playlist you want to use. This document explains how to enable and use VoiceOver

iPod shuffle (4th generation): Enabling and using VoiceOver - Apple Support