We present constraints on neutrino masses, the primordial fluctuation spectrum from inflation, and other parameters of the $Lambda$CDM model, using the one-dimensional Ly$alpha$-forest power spectrum measured by Palanque-Delabrouille et al. (2013) from SDSS-III/BOSS, complemented by Planck 2015 cosmic microwave background (CMB) data and other cosmological probes. This paper improves on the previous analysis by Palanque-Delabrouille et al. (2015) by using a more powerful set of calibrating hydrodynamical simulations that reduces uncertainties associated with resolution and box size, by adopting a more flexible set of nuisance parameters for describing the evolution of the intergalactic medium, by including additional freedom to account for systematic uncertainties, and by using Planck 2015 constraints in place of Planck 2013. Fitting Ly$alpha$ data alone leads to cosmological parameters in excellent agreement with the values derived independently from CMB data, except for a weak tension on the scalar index $n_s$. Combining BOSS Ly$alpha$ with Planck CMB constrains the sum of neutrino masses to $sum m_ u < 0.12$ eV (95% C.L.) including all identified systematic uncertainties, tighter than our previous limit (0.15 eV) and more robust. Adding Ly$alpha$ data to CMB data reduces the uncertainties on the optical depth to reionization $tau$, through the correlation of $tau$ with $sigma_8$. Similarly, correlations between cosmological parameters help in constraining the tensor-to-scalar ratio of primordial fluctuations $r$. The tension on $n_s$ can be accommodated by allowing for a running ${mathrm d}n_s/{mathrm d}ln k$. Allowing running as a free parameter in the fits does not change the limit on $sum m_ u$. We discuss possible interpretations of these results in the context of slow-roll inflation.
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