Q: Backup and restore of new Iphone 5s

Restoring the device will not remove any native iOS apps. They are part of the operating system and will be installed with it.

No, it is because you did not download them from the App store. These are available for the 5S and iOS 7 devices with new activations. You need to go into the App store on the phone and you can find them. See this support document for a little help. http://support.apple.com/kb/HT5913

You're welcome. Try to make the purchase and the price may go away. This is what I have read from others. If that doesn't work, you can always get in touch with iTunes support about it. http://www.apple.com/emea/support/itunes/contact.html