The iTunes Store app is all you should have. Its icon is purple with a white circle around two eighth notes (musical notation). That's all I ever had on my iPhone 4S, iPhone 5 and iPads (all from Verizon). Also, the iTunes store is a stock app, meaning you cannot delete the app on a stock iOS device. Anything else is likely something you installed from the App Store.
I'm sorry you no longer have itunes on your phone Smartblonde007. Have you tried reconnecting your phone to itunes for any current updates that may fix the minor problems on your phone. Performing a restore on your phone will help with the issues. Make sure you have all your data backed up and keep us posted.
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