You should keep a local copy of all of your song files, and then back up that copy in case your hard drive (or your computer) fails. Even with iTunes Match, what you get back when you download songs is not exactly like the originals. For matched songs, you get back the song file that Apple sells on iTunes Store, which may be a different format and bit rate compared to your original files imported from music CDs. For uploaded (not matched) songs, depending on the original song file format and bit rate, the version you get back may not be the same file because the song file may be converted during the upload. Your original song files (stored on your computer) are not altered by the process of creating your iCloud Music Library.
Did you have iTunes Match first (before Apple Music), when your iCloud Music Library was created?
Show your iTunes music library using the Songs view, to show a plain list with columns. Show the column for iCloud Download (if it's currently hidden), which is a narrow column with a cloud symbol in the heading. You can show/hide columns by right-clicking the heading row of song list. You can move a column left or right (in the song list) by dragging the column by its heading. Also show the columns for Kind and iCloud Status.
In the iCloud Download column, songs that are currently NOT stored in your computer's local storage have a cloud with down-arrow symbol. If that symbol is there for a song, it means that song is only stored in iCloud. Songs you want to sync to the iPod must be stored locally (no cloud with down-arrow symbol).
In the Kind column, songs that have Apple Music AAC audio file will not sync to your iPod. Other songs should sync.
In the iCloud Status column, the current iCloud-related classification of songs is shown. Songs are Purchased, Matched, Uploaded, or Apple Music. If you subscribe to (pay for) both iTunes Match and Apple Music, when you download songs from your iCloud Music Library, the songs that are Apple Music are treated as subscribed (not owned) songs. They use DRM which requires verification of your subscription when played. They will not sync to your iPod. Songs that are Purchased, Matched, or Uploaded are treated as owned songs; they should sync to your iPod.