To answer all OP's concerns:
1. You are not required to file a report at the airport. If I remember correctly, CX/KA requires that the pax must report the damage to a baggage office within 7 days.
If succeed, you will receive a confirmation in XXXCX????? or XXXKA????? (XXX is the airport office that is responsible for the claim and ????? is the reference number).
2. It is best if you can show the damage in a picture that include the baggage tag involved in the flight (so it will be difficult to argue that it is pre-existed). And unless it is extremely necessary, don't remove the tag until the claim can be finalized.
3. Proactive in your claim - CX (possibly KA as well) is really lousy in baggage claims, especially overseas offices. If you can, always call HKG office to force the local office to respond.
4. Don't trust CX on FB. At the least - a baggage claim can't be filed online (but the form is there, which it is useless until you have a reference number).
5. Based on my experience (AC/UA/CX/SQ), CX is one of the worst in processing baggage claims (My original choice was SQ. But SQ ultimately resolve the issue relatively properly than CX. So I vote for CX instead.) You will get something for sure. Don't expect a lot.
6. If you have travel insurance and you don't require CX/KA assistance in the claim, filing the claim with the company is way much easier.