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?
This question is more complicated than it seems like it should be! There is no threshold mass or density beyond which an object becomes perfectly spherical; even supermassive stars are slightly oblong. The only exception is black holes, which are perfectly round up until you reach the quantum level. If we want a simple answer, most guesses are somewhere around 1/10000 the mass of earth, or 6*1020 kg, but that is very approximate and depends on the composition of the object.