How can I reduce screen brightness below the minimum without an app?
Editing /system/build.prop and changing the ro.lcd_min_brightness property may work.  That property may not exist in your file (It doesn't in the file for my GS3), in which case it will default to 20.  Lower numbers mean lower brightness.

This doesn't actually work on all devices though.  For example, on the Galaxy S2, setting the property to a lower number will allow you to set the brightness control lower, but the actual screen brightness won't go any lower than it did before.

Alternatively, Cyanogenmod includes controls to customize the auto-brightness levels under settings > cyanogenmod settings > display > automatic backlight.  There's an explanation of how to use the custom configuration over here at rootzwiki.