No taillights

We also have this wire hanging and can't figure out where it goes

Because your post was titled tail lamps, I looked up wiring diagram for park lamps. Do the front park lamps work? The front and rear get voltage from the same place. Did you check the fuse? The ground wire at the rear is black, you need a good ground also. the brown wire off headlamp switch, is there voltage with head lamp switch on? Use a test light. If no voltage, the problem is before the test point. Headlamp switch--fuse--wiring circuit. If it is hot, you may be able to use long test wire from brown wire at headlamp switch to brown wire at rear lamp, just splice into it. If rear lamp works, something wrong with that wiring circuit. Just my opinion.

Just to add, that long wire for testing, just lay it on the ground.