You need to check fuses then if they are OK test to see if you are getting power at the switch, screen and light, if no power then you probably have a break in the harness which is usually where it leaves the car and enters the tailgate.
Yes would guess at the fuses too, this was always a big problem with Zafira's , once it was cold if you pressed the windscreen washer it would blow the fuse for front wipers and boot lock so would get the old manual out first and do a little fault finding.
Unlike the other items, the luggage compartment light is inside the boot, rather than (as the others), in the tail-gate. It is possible that the only problem with the light, is that the bulb has 'gone', so worth checking that before you try other things.
Also, there is a possibility that the earth connection to the tail gate has failed, so try an independent earth point whilst you are checking the live feeds.