You can say either 'I live in the south of China' or 'I live in southern China'. In the first, 'south' is a noun, and in the second, 'southern' is an adjective. You only put 'South' directly before a place name when it is part of the official name of that place, for example, South Africa or South Korea. It is also capitalised in these cases. You cannot say 'south China', because this is not the official name of a country or region. The same rule goes for all points of the compass and all place names.

Southern China since you are just describing the region. You would only say "South China" only if that's an official or common name of your state or region (for example, South America, South Carolina, etc.)

All three versions are correct.