Is it possible to remove the TouchWiz UI and have the phone behave like the vanilla Android UI?
It's impossible to root out TouchWiz without modifying the Samsung source code to remove and replace it; it's heavily embedded in the OS. This would be a massive effort and I don't believe anyone's done it. The only alternative is a ROM built from stock in the first place, such as Cyanogen as Ryan mentioned.
You can try installing Google Now Launcher.
The Google now Laucher will give a complete stock experience. You can check out the video here.
If the play store says that your device is not compatible, you can download the '.apk' from this article by phonearena.com
Note: This does not customize the phone application. You will need Android 4.4 if you expect the launcher to work perfectly, but it will work with earlier versions, at least partially. You probably want at least version 4.1.