What happened to particles with less energy than their rest mass? Today, particles like electrons can only be created if there is enough energy that exceed their corresponding rest-mass energy. As far as i understand, all particles were massless before the higgs field developed its non-zero vev. So my question is if particles were created during that epoch with less energy then its today rest-mass, what happened to them after the higgs field turned on? Is this even a plausible question or where the energies just after the big bang so high that all particles had enough energy? Or can it be that they somehow cease to exist, which is very unlogical.
 A: The data from observations of the cosmos are still subject of research. In the current Big Bang Model

particles  are assumed to follow the SU(3)xSU(2)xU(1)  of the standard model of particle physics,
up to the time the weak interaction symmetry breaks with the Higgs mechanism, at a time  below $10^{-10}$ seconds. Up to that time  the hypothesis is the particles are massless ( they have an invariant mass of zero).
Before the Higgs mechanism gets into action, all masses are zero. After, some remain zero rest  mass , as the gluon and the photon, and others acquire a fixed invariant mass according to the table of elementary particles.
You ask:

So my question is if particles were created during that epoch with less energy then its today rest-mass,

they were created with zero mass in a plasma environment of very high energy four vectors describing them.

what happened to them after the higgs field turned on

The ones that acquired a fixed invariant mass, took the energy from the plasma at the time of symmetry breaking.
All this in the present day model and understanding of the data and observations with mathematics. Research, observational, experimental, and theoretical is still going on in cosmology.
