Continuing on from part 1. After submitting a repair request online, I received email instructions on how to send my faulty Xbox 360 for warranty service. Unlike in the US, we have to provide our own shipping container, although the postage is free. I managed to buy a mailing box from the post office for about $2, and stuffed the Xbox 360 inside, with some bubble wrapping around the 360 to prevent it from moving around too much during transport.
So my Xbox 360 is on its way to the designated service centre. I should get an email once they have received my package, and hopefully, it will be fixed and returned to me in about a week after that.