We study ultradilute 39K Bose-Bose bulk mixtures and droplets in an external harmonic potential that confines them in one spatial direction towards the two-dimensional (2D) limit. Equations of state for several confinements are obtained with quantum Monte Carlo (QMC) at T = 0, using interaction potentials that include information on the s-wave scattering length a and the effective range reff. Performing the calculations using two different interaction potential models we have determined the range of confinements for which equations of state are universal in terms of a and reff. Based on the QMC equation of state, we develop a 2D QMC density functional for each confinement strength and use it together with the local density approximation to determine properties of the self-bound drops. For moderate squeezing, energies and droplet profiles obtained using the 2D QMC functional agree well with those obtained using 3D functionals, while offering a substantial reduction in computational cost, and a consistent approach in crossover to 2D. Noticeably, our results approach 2D mean-field (MF) + Lee-Huang-Yang (LHY) predictions only for the most strongly confined systems for which universality in terms of a and reff is observed. This implies a very narrow range of confinements for which 2D LHY functionals are applicable, which affects the study of vortices using 2D functionals.