Now THAT is interesting. In fact that is outright bizarre. Somehow you are in read only mode against a MAIL account that supports reminders.
Ok time to find out what is going on. Need a few screen shots now.
Please make sure there is no personal info showing in the screen shots.
First: The accounts under Mail, Contacts, Calendars.
Second: iCloud settings. I need the to see Mail thru Find My iPhone
IF any of the Mail accounts have the word Reminders please note when you go into that account if there is a on/off slider for Reminders.