I have an iPhone 5 myself and while a few apps do not appear under the mobile data list, nearly all do. If you can only see the native apps something is amiss with your system. The typical repair steps are forced restart, restore from backup and restore as new.
Forced Restart which apparently you have tried but try it again, please:
Hold down the Home and Wake/Sleep buttons at the same time for about 15-20 seconds until the Apple logo appears. You will not lose anything.
Restore:
First backup your device using iTunes. Then import your photos to your computer and copy all important data. Try restoring from backup first and testing it. If it does not help, you may have to restore as New and reconfigure from scratch as the backup could be corrupted. For this reason it is important to have your photos and data saved separately from the backup.
https://support.apple.com/en-us/HT201252