I suppose the SES (State Emergency Service) is the equivalent of the THW, even to the point of being set up at about the same time to help in case of a war but dealing with natural disasters and other incidents that are less dramatic. THW is federal but SES is state, but I'm sure that's trivial.
Police are of course paid.
Paramedics who ride in ambulances attending emergecies are paid - perhaps not enough. But first aid at festivals and larger events are handled by organisations like St Johns Ambulance, which are mostly volunteer-based organisations. It sounds like perhaps it is structurally different, but has the same effect.
Firefighters are divided. Metropolitan firies are usually paid, but in country towns they are volunteer.
I have noticed many similarities between Australia and Germany that seem surprising. Perhaps this is another one to add to that. Maybe it isn't so surprising though. It would seem like every country town couldn't have its own firefighting force if they have only very rare fires or if they are particularly seasonal.