This study addresses the collapse behavior of neutron star (NS) mergers expressed through the binary threshold mass M_thr for prompt black hole (BH) formation, which we determine by relativistic hydrodynamical simulations for 40 equation of state (EoS) models. M_thr can be well described by various fit formulae involving stellar parameters of nonrotating NSs. Using these relations we compute which constraints on NS radii and the tidal deformability are set by current and future merger detections revealing information about the merger product. We systematically investigate the impact of the binary mass ratio q=M_1/M_2 and assemble different fits, which make different assumptions about a-priori knowlegde. We find fit formulae for M_thr including an explicit q dependence, which are valid in a broad range of 0.7<=q<=1 and which are nearly as tight as relations for fixed mass ratios. For most EoS models except extreme cases M_thr of asymmetric mergers is equal or smaller than the one of equal-mass binaries. The impact of the binary mass asymmetry on M_thr becomes stronger with more extreme mass ratios, while M_thr is approximately constant for small deviations from q=1. We describe that a phase transition to deconfined quark matter can leave a characteristic imprint on the collapse behavior. The presence of quark matter can reduce the stability of the remnant and thus M_thr relative to a purely hadronic reference model. Comparing the threshold mass and the tidal deformability Lambda_thr of a system with M_thr can yield peculiar combinations of those two quantities, where M_thr is particularly small in relation to Lambda_thr. Hence, a combined measurement of both quantities can indicate the onset of quark deconfinement. We point out new univariate relations between M_thr and stellar properties of high-mass NSs, which can be employed for direct EoS constraints or consistency checks. (abridged)
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