First assumption: The backup key is saved on WhatsApp servers too. Otherwise a local phone to phone backup would not work?
TL:DR: No :)
The protocol is quite complicated and not limited to WhatsApp but works generally like this; The phone that uses chat app is called the first device. This device is the almighty device storing all sensitive data like private identity keys, connection passwords, local encryption keys, attachment encryption keys etc. All other devices are called second devices, this includes WhatsApp web and desktop clients.
Whenever a new device appears it needs to be authenticated via the first device first. WhatsApp does this using an initial QR-code (other apps use SMS tokens), and then via a set of proofs. Eventually the first device decides if a secondary device is allowed control over the sensitive data. If so, the second device receives these keys on request. With these keys the second device can download the contact file and decrypt it using the requested key. Same works for all media messages.
Whenever you install WhatsApp on a new phone, this entire process will take place and the first device will authorize the new phone and immediately locks out one of the instances. You then have the choice to choose one of the phones. If you choose the new phone, ownership of all secrets will be transferred to the second device, which then becomes the first device.
Apart from some OAuth and Google security this answers your second question as well.