iTunes doesn't allow changes

Have you actually selected what music you want to go onto your iPod from the new library?

In the screenshot above:

1. select the Device and then Settings/Music (1.)
2. turn on Sync Music (2.)
3. choose either Entire Music Library, or Sync selected playlists, artists, album and genres (3.)
4. If you choose the latter (sync selected...), then option 4. appears. Make your choices
5. Click on the Sync button in the bottom right corner of the window

My other problem is iTunes won't allow me to make any changes. I was trying to rearrange music in my playlists. The songs don't move.

To rearrange songs within a Playlist, you need to have the first column (the numbers column) selected, as shown below:

A Regular Playlist will probably already be in this state, but a Smart Playlist will need you to select the column first.

I also tried to move new music from "recently added" to playlists I have - nothing.

You do not "move" songs from one playlist to another; instead, you add them to another Playlist. Select the song and then click on Song/Add to Playlist. I use Windows, but the iTunes layout should be similar:

when you move the cursor over Add to Playlist, a list of all your existing Regular Playlists appears. Drop your song into the one you want. Note that you cannot drop a song into a Smart Playlist.

To recap, if you actually wish to move a song from one playlist to another, that is, you want to delete it from a particular playlist and add it to another one, simply add the song to the new Playlist and then delete it from the existing one.

You cannot delete a song from a Smart Playlist, or drop it into one. To add or remove a song from a Smart Playlist, it needs to match, or not match the rues of the Playlist.

The only thing I can pick up on from your post is:

I was referring to something like moving a newly added song to a specific playlist not being allowed.

There is nothing in iTunes that could cause this as far as I know. The only possibility is if there is a read-only restriction in the iTunes folder, where the files are located. Unfortunately, I can't offer any more help on that since you're using the Mac operating system, about which I know very little. That's why I don't partake in the Mac discussions.

Other than that, I'm not sure what (of the various points I suggested) you have checked, nor which issue you are referring to, since you haven't been specific enough.

May I suggest that you start a new discussion in the iTunes for Mac arena?

iTunes for Mac

Are you saying that none of my suggestions help with any of the three problems? That's a bit of a puzzle.

I suggest then that you try changing the Sync option to the other one. So if it's currently set to Sync Entire Library, change it to Sync selected playlists etc... and choose a few playlists and artists. Then click on the Sync button in order to ensure your new selections take effect on the iPod. That must have some effect. Once it does, change the option back to the one you want and Sync again.

As for the Playlist rearrangement issue;

1. did you manage to arrange the Playlist in the order I suggested, with the arrow at the top of the numbers column? If you did...
2. what happens when you try dragging a song to a new position, what additional icons do you see?
3. If my suggestion doesn't work, the only thing I can think of is to uninstall and then re-install your iTunes programme.

   As a part of number 3. above, which version of iTunes are you running?

By the way, I've just tried adding songs that are in the Recently Added section to another playlist. I clicked on the album, to reveal the tracks, then simply dragged my choice of song to the Playlist I want. What happens when you try this?

Do you see any error messages (or icons) when you try any of these things?