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On-off intermittency occurs in nonequilibrium physical systems close to bifurcation points and is characterised by an aperiodic switching between a large-amplitude "on" state and a small-amplitude "off" state. Lévy on-off intermittency is a recently introduced generalisation of on-off intermittency to multiplicative Lévy noise, which depends on a stability parameter $α$ and a skewness parameter $β$. Here, we derive two novel results on Lévy on-off intermittency by leveraging known exact results on the first-passage time statistics of Lévy flights. First, we compute anomalous critical exponents explicitly as a function of arbitrary Lévy noise parameters $(α,β)$ for the first time, by a heuristic method, complementing previous results. The predictions are verified using numerical solutions of the fractional Fokker-Planck equation. Second, we derive the power spectrum $S(f)$ of Lévy on-off intermittency and show that it displays a power law $S(f)\propto f^κ$ at low frequencies $f$, where $κ\in (-1,0)$ depends on the noise parameters $α,β$. An explicit expression for $κ$ is obtained in terms of $(α,β)$. The predictions are verified using long time series realisations of Lévy on-off intermittency. Our findings help shed light on instabilities subject to non-equilibrium, power-law-distributed fluctuations, emphasizing that their properties can differ starkly from the case of Gaussian fluctuations.