For what you asked first, here is what you get first:
DeTune is an App that helps you get Songs from your iPod.
For everything else regarding Organization:
Open iTunes and hit CMD+,
You should see the Settings, if not go to The Menu and click iTunes -> Preferences.
Here is all I know after I struggled with iTunes (a lot):
You can specify Import Settings on the first page of the iTunes Settings page.
At the Bottom you can also say that you want to download the names of the tracks.
Regarding the Organization :
iTunes handles the media the way you specify it in the Advanced Tab inside the preferences.
So you should check on that via going to preferences -> Advanced.
You should see the place where all the media is stored.
Also you should see checkboxes for the automatic management and the option to copy the media inside the iTunes Media Folder.
If you want make iTunes do the work, check both.
And well, that is basically all you can do to achieve what you want.
To better organize your media with iPhones/iPods there is no way around iTunes. It's simply the only App that can handle your device completely.
If you want to organize your media professionally, tag them manually or via great tools like Picard (from MusicBrainz), Metadatics, Tagger or something else. Organize it in Folder you like inside your Music Folder and don't let iTunes organize them. Then (if tagged correctly) drop them into iTunes.
To be sure you added the files, maybe add a Finder Tag to remember yourself in the future.
I had exactly the same issue, it's commonly called "ghost songs" and it still exists in iOS 10.x There was no way to remove these songs using iTunes (I could deleted them there but on next sync they were just back) or any delete function on the phone itself (I deleted all music on the phone, but iTunes kept showing the ghost songs while the phone said there are no songs on the phone).
Here is how I was able to fix it:
After that the ghost songs were gone.
Check your Restrictions.
Go to Settings > General > Restrictions.
Enter your passcode.
Look under "Allowed Content" for "Music, Podcasts & iTunes U", it should say "Explicit" to the right.
If it does not, tap on it and enable it.
If everything is greyed out when you arrive at Restrictions, it means they are disabled; I've read people have had some success enabling them, allowing explicit content and then disabling Restrictions again.