Is it true that an ultrasound transducer has a fixed center frequency that it HAS to be driven near or at?

They are designed to work in efficiently over particular frequency range.

If you use a lower frequency, the wavelength of the sound is too long compared with the size of the transducer and you get poor efficiency in transferring energy from the transducer to the fluid, like trying to reproduce low audio frequencies with a small loudspeaker.

It you increase the frequency, the inertia of the fluid and the transducer limits the amplitude. The peak acceleration of a sine-wave signal is proportional to frequency squared, so going from 20 kHz to 60 kHz means 9 times the acceleration (and therefore 9 times the force) to get the same amplitude of vibration.

Piezo transducers are basically voltage-controlled devices, so you are unlikely to burn them out unless you exceed the voltage rating. Beware that they do not have a pure resistive load, so they can form part of an electrical resonant circuit, either intentionally (e.g. a quartz crystal oscillator) or by accident. They mechanical part of the system can also have resonances that are beneficial or not, depending on the application.