What is the minimum mass required so that objects become spherical due to its own gravity? Is this minimum mass known? or maybe, is it given in terms of density? If so, how much density is the minimum to have an spherical object due to its own gravity?
Icy objects, such as most in the Kuiper belt can reach an equilibrium if they are about 400km across, whereas the rocky asteroid Pallas, at 572km clearly has an irregular, non spherical shape. All rocky objects larger than Pallas (and there aren't many) are spherical.

Rock tends to be stronger than ice. Rocky objects are able to withstand their own gravity for longer than icy ones. Pallas is a reasonable cut of point. The next smaller asteroids (Vesta, Hygiea etc) are round-ish, but not in hydrostatic equilibrium. On the other hand Small, icy moons such as Miranda and Mimas are in, or close to equilibrium. Mimas has a diameter of just under 400km.