Let's see whether we can address at least some of the issues.
I'm a Windows user, so I'm not an expert on the Mac by any means, but if I add to what Zurarczux has told you, you can hopefully make progress.
From what I understand there are separate apps on your Mac for Music, Podcasts and Audiobooks (all classed as Media). Those apps effectively allow you to save media to your Mac, as well as managing them, before adding them to your iPod. To add them to the iPod, you will use the Mac's built-in Finder.
So somewhere in Finder there should presumably be an option to choose which Podcasts and Audiobooks (that are already on your Mac) to add to the iPod, in a similar way to the music.
One thing about Podcasts; a "Podcast" is media that has been downloaded "as a Podcast", through a Podcatcher, not simply media that the originator describes as a podcast. Look at the screenshots below for a proper Podcast, downloaded by iTunes, as a Podcast:

The screenshot above show the Options tab (1.) for a Podcast. The podcast series is a discussion about is about music, artist and songs, so the media kind (2.) is set to Podcast. It's this tag (field) that determines whether the media is "a Podcast" or not.
But here's the issue, which you will need to check on your "podcasts":
when we look at the Details tab (3.) for the same episode, the information is significantly different:

Note that the genre (4.) is listed as "podcast". The genre is most definitely is not "podcast".
The word genre is defined by a simple Google search as a category or style of art, literature, music, or film that is recognized by a specific set of shared characteristics, themes, or rules. Since the audio is a discussion about Kate Bush's song Wuthering Heights, one might consider the genre to be:
- music (although I wouldn't use it)
- spoken word (also not my first choice)
- discussion about music
or a term that describes the series or even just that episode. What is most definitely not is "a podcast". If the discussion was about podcasting, then I would agree that the genre could be "Podcast".
Why am I being pedantic (passionate) about this? Because this mistaken belief leads people to think that the genre determines whether or not the episode will be added to your iPod as a podcast, when it does not. The media kind determines whether the file (track) is a podcast and it is the media kind field that will allocate the media to the Podcasts section of your iPod Classic.
So check your "podcasts". The media kind must be set to Podcast in order for them to be in the Podcasts section of your iPod. That will also place the episodes in a different part of your music library. Note also that some of the tags are either different, or used differently, in a podcast compared to music.
It's a similar situation for Audiobooks. Once again, the media kind determines whether or not your iPod puts the tracks in the Audiobooks section of your iPod. And here's a crucial point to consider: the title of a piece in music may not be used as the title in an audiobook or podcast. So if you change the media kind from music to podcast or audiobook (or any other combination), you may have to search through your library in order to make sure you know what the series title and episode titles are.
To put it another way:
- in music, you have song title, album title and artist name
- in a podcast you have episode title, series name (podcast field in the Details tab) and author field (on the Details tab) which is the name of the people who created the podcast
- in an audiobook, you have title, chapter and author (and note that effectively, an audiobook's "author" would be a song's artist
These fields (tags) do not smoothly work when switching between media kind. But there's little that can be done about it. Confusing, isn't it?
If you would like more information about how one downloads Podcasts properly (so that they are podcasts), let me know. I'll hazard a guess now, that the Podcast app on your Mac is the correct place to find and download podcasts. Downloading a "podcast" by any other method is actually probably downloading an audio file in mp3 format.
Regarding "Notes" on an iPod Classic:
the iPod Classic's note feature is very, very basic. I never used it for notes, it simply wasn't sophisticated enough. You cannot compare the Notes feature on an iPod Classic with the most basic notetaker on an iPod Touch (or iPhone), or on an Android phone.