IMac has begun to require computer's password when coming out of "sleep" mode

In addition to the previous post, check in System Preferences->Security & Privacy->Advanced and make sure that “logout after X minutes of inactivity” is NOT checked.

 1 - Suggest restarting in Safe Mode. This will perform a Disk Repair, clear cache files and only load Apple Software, extensions and fonts. The boot up will be slow and can take some time - Normal.

2 - Sometimes a Safe Boot followed by a Normal Boot will just put things right.

3 - Does the issue present in this mode ?

4 - If not - there could be something in the main User Account playing up. To further isolate this - Set up users, guests, and groups on Mac. Then log out of the Main User account and log into the dummy account and test again if the issue persists.

5 - If the issue is present in the dummy account - then, this appears to be a System Wide issue on the computer.

6 - If after performing each of the above steps is the order that have been presents  and still have issues - please advise for possible further assistance.

Look at the bottom right of the window in the screenshot you posted. There is a button “Advanced…”.

Click the padlock, authenticate, and the button becomes active.