With shiitake mushrooms after the first flush the logs or bags are "dunked" which means they are submerged in clean
water after the first flush to re-hydrate and to promote further flushes. Basically they are weighted down over night. Mycelium is very buoyant so they have to be held down with something.
Oysters are definitely less picky about temps and
co2 levels so they are a very good first mushroom variety to grow.
If you can get
enough cardboard spawn together you could make a bed outdoor and fill it with pasteurized
straw or
wood chips.
Pine is no good, neither are most conifers. Eucalyptus is probably no good do to is oil content, that might go for gums. Maybe certain acacia or wattles like black acacia would work OK. Generally hardwoods like aspen, oak, poplar cottonwood...broad leaf deciduous hardwoods are what you want for mushroom cultivation. I know the flora and fauna of Australia are pretty much the most unique in the world so its hard for me to offer any advice on stuff that might grow in your region due to my ignorance. I live in California and we do have Australian
trees, 2 eucalyptus varieties that I know of and some acacia that have become invasive here. If you have acacia near you that is not resinous that might be your best bet.