Not everywhere in Australia, you don't have to fit them but your home needs to meet a mandated minimum '6 star' energy rating. This comes down to everything in the house such as wattage per square meter of lighting, double glazing in windows etc.

The biggest pitfalls here usually with solar hot water is the storage capacity, especially with a family.

Americans are afraid of the "third-world look" of solar water heaters they know from vacations in Europe

I don't think this is true? Also for what it's worth I've spent time on the roofs of a lot of apartment buildings in the bay area and seen many solar water heaters. Newer buildings and homes tend to just have solar panels if anything, though.

I guess I'm not privileged to have seen that and would mind a setup on my house. I'm in AZ