hey bd,your contents policy should cover the damage done to the carpet as it was damage done at a single instant in time and not gradual damage. However for a successful claim, the tanks must be covered by the insurance.
Each and every insurance policy has what's called an earthquake levy. Basically if you have to make a claim at the moment, you first call the EQC, lodge your claim and then your insurance company.
The one claim per dwelling will refer to home owners (house) insurance. It will be separate to your contents policy and if you were making a claim, you'll need to lodge one with the EQC and your own insurance provider.
Carpets can be a tricky one. If the carpets are attached to the side of the house and will not come undone if the house was turned upside-down, the home owners house policy will cover them. If the carpets are like rugs that are untacked; they are classified as contents. If the carpet was damaged by a content, in most cases the contents policy will cover it, but the EQ is an exceptional circumstance and it is likely to be replaced under the house policy.
Fridge / freezers - It's best to take pics of what you intend to claim on, but since that is not always possible you can try making a claim and leaving it in their hands. Keep your receipts from items you had to purchase immediately after the quake and you might be able to make a claim based on that as those items may be viewed as replacements for what was immediately lost.