Using density functional theory calculations (both perturbed and unperturbed) as well as thermodynamic and ballistic transport equations, what follows investigates thermal and mechanical properties of two-dimensional boron monolayers (δ6, α, δ5, and χ3 sheets with respective vacancy densities η=0, 1/9, 1/7, 1/5) as they relate to the vacancy density. The triangular (δ6) sheet's room-temperature phonon and electron thermal conductances are found, respectively, to be roughly 2.06 times and 6.60 times greater than those of graphene. The Young's moduli, calculated from longitudinal and transverse sound velocities are in good agreement with those obtained from elastic constants. Values range from 171 to 619 N/m, two of which (619 N/m for α sheet and 546 N/m for δ5 sheet) exceed graphene's Young's modulus (∼340N/m). It is determined that the vacancy density has a diminishing effect on both the phonon heat capacity at constant volume and the phonon ballistic thermal conductance, but no regular correlation on the electron heat capacity and electron ballistic thermal conductance.