We characterize, by means of large-scale fermion quantum Monte Carlo simulations, metallic and deconfined quantum phase transitions in a bilayer honeycomb model in terms of their quantum critical and finite-temperature properties. The model features three distinct phases at zero temperature as a function of interaction strength. At weak interaction, a fully symmetric Dirac semimetal state is realized. At intermediate and strong interaction, respectively, two long-range-ordered phases, each breaking different symmetries are stabilized. The ordered phases feature partial and full gap openings in the fermion spectrum, respectively. We clarify the symmetries of the different zero-temperature phases and the symmetry-breaking patterns across the two quantum phase transitions between them. The first transition between the disordered and long-range-ordered semimetallic phases has previously been argued to be described by the (2+1)-dimensional Gross-Neveu-SO(3) field theory. By performing simulations with an improved symmetric Trotter decomposition, we further substantiate this claim by computing the critical exponents 1/ν, ηφ, and ηψ, which turn out to be consistent with the field-theoretical expectation within numerical and analytical uncertainties. The second transition between the two long-range-ordered phases has previously been proposed as a possible instance of a metallic deconfined quantum critical point. We further develop this scenario by analyzing the spectral functions in the single-particle, particle-hole, and particle-particle channels. Our results indicate gapless excitations with a unique velocity, supporting the emergence of Lorentz symmetry at criticality. We also compute the finite-temperature phase boundaries of the ordered states above the fully gapped state at large interaction. The phase boundary vanishes smoothly in the vicinity of the putative metallic deconfined quantum critical point, in agreement with the expectation for a continuous or weakly first-order transition.