It might be worth talking to your agent about. They may well say the owners will pay to have it done for you. Professional spraying should help a lot with your insect pests, but as others have said it can be exey.
I have to admit, spiders don't bother me a great deal - we get heaps of little ones around the place and I kind of like them because they eat the smaller, bitey insects I wish they would catch some of our moths that we have around the joint atm though.
Not much you can do about possums (even trapping and re-location doesn't work because you can't take them too far away so they just end up coming back again), and IME mice are extremely difficult to get rid of, particularly if you have lots of vegetation around your home. If the carpet is old and a bit yucko then that will attract the mice, so it could be worth asking for new carpet.
ETA: The ants - we have been getting these tiny ones lately - you can hardly see them but they are attracted even if I put a knife in the sink still with some butter on from my toast! I have never seen them in Syd before. My sister had them up in Cairns but they seem to have found their way into my kitchen!