A qubit may undergo Landau-Zener transitions due to its coupling to one or several quantum harmonic oscillators. First we show that for a qubit coupled to one oscillator, Landau-Zener transitions can be used for single-photon generation and for the controllable creation of qubit-oscillator entanglement, with state-of-the-art circuit QED as a promising realization. Second, for a qubit coupled to two cavities, we show that Landau-Zener sweeps of the qubit are well suited for the robust creation of entangled cavity states, in particular symmetric Bell states, with the qubit acting as the entanglement mediator. Finally, for a qubit coupled to an environment or bath we propose to employ dissipative Landau-Zener sweeps of the qubit for the detection of bath properties. At the heart of our proposals lies the calculation of the exact Landau-Zener transition probability for the qubit, by summing all orders of the corresponding series in time-dependent perturbation theory.