The password for the iMac has changed by itself

Do a backup, preferable 2 separate ones on 2 drives. Boot to the Recovery Volume (command - R on a restart or hold down the option/alt key during a restart) and select Recovery Volume. Run Disk Utility and select First Aid. Then re-install the OS.

OS X Recovery

OS X Recovery (2)

Did you try restarting with the command - R keys held down? The option/alt during a restart doesn't work with Sierra. From the article I posted earlier:

Hold down Command-R or one of the other macOS Recovery key combinations on your keyboard immediately after pressing the power button to turn on your Mac, or immediately after your Mac begins to restart. Continue holding until you see the Apple logo or a spinning globe. Startup is complete when you see the utilities window:

Reinstalling the OS shouldn't affect other data.

Get an external drive with no data on it. Try booting into the Recovery Partition (command - R on a restart) and select the old drive in Disk Utility, then Edit/Restore and select your Internal Drive in Restore from. Select the new drive as the destination. This will likely reformat the external drive erasing all data.

If you have a recent backup, use Disk Utility to erase/reformat the drive. Choose Mac OS Extended (Journaled) with Option GUID. Reinstall the OS and then use Setup Assistant to restore from your backup.

If you don't have a backup, try this before erasing Get an external drive with no data on it. Try booting into the Recovery Partition (command - R on a restart) and select the old drive in Disk Utility, then Edit/Restore and select your Internal Drive in Restore from. Select the new drive as the destination. This will likely reformat the external drive erasing all data. If that work, then erase and reinstall.

Are you booted into the Recovery Partition? It looks like you downloaded the install after a normal boot.

If you click Reinstall macOS, you get the screen shot you posted earlier?