Zero point energy would be infinite because time becomes infinitesimal. It must be finite. Because it manifests in particles it is quantized. When they appear they do not move. They are not relativistic.

More imprortant when do virtual particles spring into existence? It must be totally random statistically with heavy particles appearing less often and the electron most often.

Different particles are differently charged. Some do not experience certain forces at all. They are uncharged. These different virtual particles would have different effects on matter depending on their forces. Like charged virtual and matter particles will only effect each other.

If these virtual particles effect each other could they come together as virtual hydrogen the simplest virtual atom?

Virtual paticles have virtual waves.