Yes you can hire cars with car seats just make sure when you book your car you tell them what age your bub is, where you want the seat (behind passenger seat, middle etc) and forward or rear facing. Also you can take the port-a-cot and your bags and your stroller (we always do). We always fly with Qantas as DH is a frequent flyer but I wouldn't think it should matter.
Have a wonderful time, take lots of photos and enjoy the time with your Dad!