“battered”, in part III, means damaged.

Correct, although battered in this context has more to do with a place that suffers through natural disasters and has to endure severe weather conditions, one of the definitions for batetred is indeed "damaged and striken"

correto: battered: broken-down,crumbling,,damaged,decaying,decrepit,derelict,dingy,impaired,, injured,neglected,ramshackle,rickety, run-down,seedy,shabby,unkempt, danificado.