  Kitaev's 0-$\pi $ qubit encodes quantum information in two protected,
near-degenerate states of a superconducting quantum circuit. In a recent work,
we have shown that the coherence times of a realistic 0-$\pi $ device can
surpass that of today's best superconducting qubits [Groszkowski et al., New
Journal of Physics 20 043053 (2018)]. Here we address controllability of the
0-$\pi $ qubit. Specifically, we investigate the potential for dispersive
control and readout, and introduce a new, fast and high-fidelity single-qubit
gate that can interpolate smoothly between logical X and Z. We characterize the
action of this gate using a multi-level treatment of the device, and analyze
the impact of circuit element disorder and deviations in control and circuit
parameters from their optimal values. Furthermore, we propose a cooling scheme
to decrease the photon shot-noise dephasing rate, which we previously found to
limit the coherence times of 0-$\pi $ devices within reach of current
experiments. Using this approach, we predict coherence time enhancements
between one and three orders of magnitude, depending on parameter regime.
