Let's see one by one:
If I understood correctly, the car should be off when I do the test?
Yes, it's strongly recommended in this case.
shall I disconnect the battery?
It's not necessary. Until the ignition and the parking / headlights are turned off no voltage
present on the cluster's connections.
I should get the 'bip' sound of continuity. My multimeter measures KΩ.
Is that a digital multimeter or an analogue one?
Using analogue multimeter is not recommended on circuits with semiconductors (like the cluster).
In case of digital multimeter connect the black multimeter lead to the black wire's terminal
while use the red multimeter lead to test the fixing nuts.
can I connect the new wire on the black wire there, which in turn should be used as the new ground just to check if the problem goes away?
Yes, that works also.
Something else, how can I check if the cluster gets the appropriate voltage?
For this test the battery voltage need to be maintained as constant as possible,
which requires running engine or a battery charger (with 10A or more charging current)
connected across the battery terminals.
Is there a pink wire in the red connector?