Answers are to be found in rare and expensive books and research papers. Piezo device modeling can be horribly complex.
They can be looked at in slightly simpler terms, say impedance and output displacement, in which case the characteristic graphs often look like a mountain range. This doesn't account for electromechanical feedback, drift, tempco, ageing etc, but is a good start to understanding piezo characteristics and their inherent self-resonant nature. So yes, they're very frequency dependent, and driving a piezo device at too high a frequency will fry it. Always refer to the piezo datasheet, as they're all different.
Piezo materials can range from materials where the Q is so low as to be irrelevant (such as bone and Rochelle Salts), to materials where the Q is all-important, such as quartz crystals, and may approach values of 100,000 or more.
Resonance, therefore, is often associated with piezo materials, but not necessarily so. A Piezo material is any material that generates and electric potential under stress.