I am currently in the process of renewing my British passport while living in Australia. Once it's been renewed, will it have a copy of my permanent residency visa? Or does it just stay in my expired one and I get that back?
The Passport Office will not do anything to the visa in the old passport. All they will do is slice the corner off the photo page and cover to indicate that the passport is cancelled. You will get this passport back, with the visa intact (no corner cut off). In most circumstances, said visa is valid for travel in conjunction with the new passport, but that's another question.
No. Australian visas are now completely electronic and linked to your passport number. Visa labels are obsolete and no longer necessary.
When you get your new passport, you will need to advise the Department of your new passport details so that the visa can be transferred across electronically. You will not get a new label.