Could there be a generation 0 of elementary particles (that we couldn't find yet)? It appears currently unlikely that a fourth generation exists, but would a "generation 0", lower in mass and longer lived, potentially exist?
 A: An interesting question, but the answer is "no" unless one stretches the definition of "generation" far beyond its conventional Standard Model use.
As noted in J.G.'s comment, any lighter generation with significant coupling to the familiar 3 generations would be seen in the weak decays of the known quarks and leptons. For example, the electron and proton would not be stable and our universe would be very different.
One can get around this limit if there is negligible weak coupling between the 0th generation and the known 3 generations. In other words, if its cross-generation entries in an extended 4-generation CKM matrix were all zero.
There are, however, even bigger problems with a light 0th generation.
A conventional light generation would very noticeably increase the W and Z bosons decay widths.  For example, as noted in the answer to Light neutrino number from the invisible decay width of Z
boson, and precluding heavy neutrinos as dark matter candidate, there cannot be more than 3 generations of light Standard Model neutrinos.
Finally, the 0th generation also could not have the standard electric or colour charges, or it would be copiously produced and observed in photon interactions, electron-positron annihilations, hadronic interactions, and (as noted by J.G.) in the Big Bang.
So, a light 0th generation could not couple to the Standard Model Strong or Electroweak forces, which means it could not be a Standard Model generation.
What could exist, however, are light Mirror Matter generations. These would be fermions that could have the same generation structure as in the Standard Model, but with their own independent interactions and associated bosons. They only connect to regular matter through gravity and neutral particle mixing, which makes them very hard to detect. Mirror matter is one of many possible candidates for dark matter.
