A theory about the "mystery" songs... On the iPhone's Music settings screen in iTunes (shown in your first screenshot), scroll all the way to the bottom. Do you have a list there for Manually Added Songs. Since iTunes 11 (I think) you can actually add songs manually to the device (by dragging and dropping), even if it set to use automatic syncing. I don't usually mention this in my comments, because it gets too confusing. But, if you dragged some songs to the iPhone (with Sync Music enabled), those manually added songs appear on that Manually Added Songs list (where you can manually remove them). You won't see that list there, if you don't have any manually added songs on the iPhone.
I have a suggestion for a better way to set up automatic syncing, especially since the number of synced songs is relatively low. You are currently using a genre to sync songs. Most users don't use genre to define syncing. You should use playlists instead, because they are easier to update.
In your iTunes library, create one or more playlists with songs you want on the iPhone. I'll explain this with only ONE playlist, which you can call iPhone Songs. Add all the songs you want on the iPhone to this playlist.
On the iPhone's Music settings screen, set it up as shown in your screenshots, except uncheck Workout under Genres. Nothing should be checked on the four list. Then, under Playlists, find the iPhone Songs playlist you created. Select (checkmark) it. Click Apply to sync your selection (songs and the playlist) to the iPhone.
Going forward, to update songs on the iPhone, just update that iPhone Songs playlist in your iTunes library (add/remove songs). The iPhone does not need to be connected. The next time you connect it (or click Sync if already connected), iTunes automatically updates the iPhone with the same changes. You can create more than one playlist for your iPhone. For example, one can be for your Workout. Another can be your Favorites. And so on... Then select those playlists on the iPhone's Music settings screen. After syncing, the songs and playlists are on your iPhone. You can play the songs directly or from the playlists.
This "Music" pane on R can't be edited from this view...
When you are using Sync Music setting (automatic syncing), you cannot manually remove songs that are synced automatically. That's why that list is "see only." If your iPod is set to Manually manage music and videos, THEN you can manually add AND remove songs.