The black wire goes directly to ground, attached to nothing else with the switch OFF.
When you click the roller switch to turn ON the dome lamps, it puts the ground connection out to the white wire which goes to the dome lamp circuit.
That circuitry and wiring has absolutely nothing to do with dash lamps however it is a separate switch part of the roller.
The brown will be at +12 with the headlights and parking lights ON.
Green is only the output of the dimmer control and goes down to the fuse panel and feeds the 5A INST/LPS fuse there.
There are two fuses for the dome lamps, the parking lamps 20A fuse and then through the headlamp switch then through the dimmer, then through the 5A fuse out to the lamps on the gray wires in the dash.
If you get +12 on the brown wire the 20A parking lamps fuse is good if not check that one.
If the dark green wire has +12 with the dimmer on HIGH then check the LPS terminal in the fuse panel and check the 5A fuse there.
If you wish to check the lamps in the dash I would use a 5A fused wire to the battery and pull the 5A fuse in the fuse panel.
Try putting the 5A fused external wire to the LPS terminal in the fuse panel to check the lighting.
PULL THAT 5A FUSE FIRST so you don't backfeed any other circuits if there is a short somewhere or problem with that wiring.
Gray wires are out of the fuse panel after the 5A fuse, and run throughout the dash area to illuminate various bulbs such as shifter, dash lamps, ashtry lights, radio day/night lighting etc.