Q: Why does amachine ask for a password when it comes out of sleep or after the screen saver has been on for 15 minutes, even though I turned off the option that requires a password when exiting sleep?
A: If you have configured some Exchange accounts using the Windows 8 built-in mail application, it will enforce any Exchange ActiveSync (EAS) policies that are required for the mailbox (Outlook does NOT do this as its not part of the OS). One of those EAS policies would have stipulated the need to enter your password after 15 minutes of idle time or exiting sleep.
Typically, you can configure this via the Users area of the PC settings (selecting the Settings charm on the Start screen and selecting the Change PC Settings link) as shown below. Notice the button to Change if a password is needed when waking the machine which toggles between requiring and not requiring the password.
If you have a policy enforcing this you will see the following:
The only way to solve this is to remove that mail account from the built-in mail application (which is what requires the policy) and then reset the ActiveSync policies (see my FAQ "Q: I configured some Exchange mailboxes then later removed the accounts, but the policies are still in effect--how do I reset those Exchange ActiveSync policies applied to my Windows 8 machine?").