RE: Can NOT get rid of music on iTunes??!
I do sync my music and also had songs syncing even though I didn't want them. What I found out was that songs I had purchased on my iPhone would default to being selected to sync on the albums view. What I would try, assuming the songs you are frustrated over are ones you may have purchased (free songs or songs of the week) but dont want, is the following.
1. Select sync music. This will give you the ability to see if anything if set to sync by default by genre, artist, album etc.
2. Choose "Selected playlists, artists, albums, genres".
3. Go to album section and ensure nothing is selected. Scroll all the way through. This is where I found songs that I purchased as set to sync just because I "bought" them on my iPhone
4. Make sure nothing else is selected to sync.
5. Sync. This is just to remove the music from your phone.
6. Unselect "sync music"
7. Sync phone
This is my theory. If you've not purchased music on your phone probably not worth trying