How can it be said that the real beginning of the real universe involved a "singularity of zero size and infinite density of mass" when no space existed at all to define any density and when nothing/zero is only an imaginary concept and not real?
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The cosmological data from observations have been fitted with the Big Bang model. Fitted means that a mathematical formula is used which fits the data with acceptable errors.
One has to be clear that mathematics is not physics. That the formula fitting the observations has a singularity at the beginning of its functional form, does not mean that physical values also follow up to that singularity. There are no data for that extrapolation , the (0,0,0,0) of the formula, and no known way to get at the data, so there is no contradiction as far as physics goes. The mathematical formulae used in physics have a region of validity, i.e. have been validated, and the origin is outside the region of validity of the present model.
With the above in mind
How can it be said that the real beginning of the real universe involved a "singularity of zero size and infinite density of mass"
the zero size is a mathematical concept, the density of mass is a physics concept. It is wrong to extrapolate a physics concept mathematically to regions where the mathematical model has not been validated, as is the beginning of the universe.
when no space existed at all to define any density and when nothing/zero is only an imaginary concept and not real?
by imaginary you mean mathematical, and by real you mean a measurable physics quantity. So it has no meaning to extrapolate physics to the mathematical singularity.
This said there is on going research for quantizing gravity and this means that the solutions at the extremes of the origin will be different, as it is different for the classical coulomb case between charged particles which has a 1/r singularity for opposite charges. Quantum mechanical solutions solve this conundrum and it may be the same for the origin of the Big Bang.
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