Setting up Payment and Billing Information

I'm sorry this is happening to you but it shouldnt be an issue. Everything you use to purchase on an xbox 360 console should be directly linked to your microsoft account. If that is in the correct region NZ and you have payment options
already connected to this account, this is the same payment option provided for your xbox 360. Of course you can try to add a payment option through your xbox 360 but its the same thing as logging into your microsoft account on a PC and adding a payment option
there. It is synonymous.

For this you would need to contact support and they can help you. 0508 555 592 or
http://support.xbox.com/contact-us

Unfortunately if there's a past due it'll have to be paid or your account goes into a temporary lockdown (until it's paid).

Subscriptions do end on their own if they're not set up as "auto renewal", you can find information on how to change your payment methods here

https://support.xbox.com/help/subscriptions-billing/billing-payment-updates/manage-payment-options-and-billing-address

First place for you to check that everything is set up properly with your account and payment methods is on your
Microsoft Account page
under Payments & Billing. Verify that your methods are not expired, up to date, have the proper information, etc and see if that's causing any issues. If there are other issues around payments and billing it should notify you there as well.

It sounds like your billing contact information is just set up as your parent's information. up at the top of the page, click on My Account, then scroll toward the bottom. On the left hand side, you'll see a column labeled payments and billing. Click on
Billing Contact Information - it will ask you to verify the account if you have not set this as a trusted PC. Here is where you can edit the account details.

Phone support can also adjust everything except the name.