We experimentally optimize the frequency of flux-tunable couplers in a superconducting quantum processor to minimize the impact of spectator transmons during quantum operations (single-qubitgates, two-qubit gates and readout) on other transmons. We adapt a popular transmon-like tunable-coupling element, achieving high-fidelity, low-leakage controlled-Z gates with unipolar, fast-adiabatic pulsing only on the coupler. We demonstrate the ability of the tunable coupler to null residual ZZ coupling as well as exchange couplings in the one- and two-excitation manifolds. However, the nulling of these coherent interactions is not simultaneous, prompting the exploration of tradeoffs. We present experiments pinpointing spectator effects on specific quantum operations. We also study the combined effect on the three types of operations using repeated quantum parity measurements.