PID vs Bang-Bang - which one is more stressing on components?
So I was wondering, do the faster cycles cause more stress on the components


Probably not. In fact, switching faster may stress component less.

Rationale: when switching slow, components have time to heat up and cool down on each cycle, which causes thermal expansion and contraction cycles and mechanical stress. Switching much faster keeps components at a constant (average) temperature and avoids thermal cycling.

7Hz is still quite slow, you'll still have some thermal cycling, but less than with the bang-bang regulation.

The noise you hear could either be due to the power supply and transformer's noise being modulated by the 7Hz PWM, or a ceramic capacitor acting like a piezo speaker, or simply from the wires. Variable current in wires causes a magnetic field which can make move very slightly. Since your current is pretty high this could be audible. Nothing to worry about though.