Follow-up to this: Can I buy two return tickets on my app and use them for each of us, i.e. can I activate two passes on the inbound and return?
I haven't tried myself but any version after 4.5.0 supports mutliple or even group travel. You can in any case email + print the tickets. Personally I don't bother, I regard the 50p surcharge on a contactless card as the modest surcharge for a flexible ticket.
I have often used the bus service (for similar reasons to you as it happens) for this hotel. From the airport you need to go down Pitt Street to get to Bishop Lane, but to the airport you're just going around the unglamorous back of the hotel to get to Bishop Lane and the bus stop isn't far off.
The 05:00 is fine, gives you are good margin. So long as you have your boarding pass for the flight you could easily make the flight at 05:15 you ended up taking a taxi if the bus doesn't show up (very unlikely scenario). It may be better to get a later flight so you can enjoy the breakfast at this hotel which is reasonably good. For Diamonds they may give you a doggy bag with a bottle of water and an apple in it at the end of breakfast, which was once unbelievably useful.