Hey there AD_77,
It sounds like your Mail, Contacts, and Calendars options are grayed out. It seems to me that Restrictions are enabled and would check to see if Accounts are set to Allow or Dont Allow Changes in your settings.
Go to Settings > General > Restrictions
From: iPhone User Guide
http://help.apple.com/iphone/7/#/iph9b66575d5
Thank you for using Apple Support Communities.
All the very best,
Sterling