But I want ask; does the relic density of dark matter have anything to do with the big bang? I mean, were these created after the big bang?

The dark matter assumption fits in the Standard Model of cosmology, which relies itself on the big bang. So in some way, it is connected to the big bang.

Another question - is the relic density of dark matter quantifiable in some kind of units, and can they be measured?

Usually, this quantity is given as the ratio of the energy density of dark matter to the critical energy density of the universe (which is 1 for a flat universe). This is the standard way to quote it.