Prior to the electroweak force splitting, did charged leptons still carry a charge of $e$? I was under the impression that, since electromagnetism didn't exist in its current state prior to electroweak force splitting that particles couldn't quite carry it prior. Would this indicate that electrons were neutral during the electroweak epoch, or did they carry a completely different charge?
Would this indicate that electrons were neutral..., or did they carry a completely different charge?
This part of the question refers to "electrons," and that might already be too presumptuous. Our current understanding of particle physics is based on quantum fields, which can be manifest in a variety of ways. Particles are just one manifestation of the quantum fields. The pattern of charges associated with the fields is part of the definition of the model (this is clarified below), so this pattern of charges doesn't depend on state-specific parameters like density or temperature. However, the properties of particle-like phenomena can depend on state-specific parameters like density and temperature. I don't have a clear picture of what things were like prior to the electroweak symmetry breaking (EWSB) transition (papers like  make me question my naive guesses), but I wouldn't be surprised if the spectrum of particles is completely different, if there even is any spectrum of "particles" at all. Here's why:
The electromagnetic and weak interactions are two different mixtures of two more-fundamental gauge fields that are often denoted $SU(2)_L$ and $U(1)_Y$.
Only the left-handed matter fields couple directly to the $SU(2)_L$ gauge field.
Both left- and right-handed fields couple directly to the $U(1)_Y$ gauge field. Their charges with respect to this interaction are called hypercharges. The left- and right-handed fields have different hypercharges.
Below the EWSB transition temperature, the electromagnetic field emerges as one mixture of the $SU(2)_L$ and $U(1)_Y$ gauge fields. Above the transition temperature, the picture is different, and so the OP is rightly questioning the idea of "electric charge" at higher temperatures. Electric charge is due to a special combination of $SU(2)_L$ and $U(1)_Y$ charges, a combination that emerges only after EWSB. However, without EWSB (for example, if the Higgs field were artificially deleted from the Standard Model), the left- and right-handed components of the fermion fields would also remain unmixed. This makes me think that not only does the idea of electric charge become dubious, but even the idea of an electron becomes dubious, because the electron is a combination of left- and right-handed fields. (I'm talking about chirality here, not helicity; the difference is emphasized in .) Without EWSB, the left- and right-handed parts might not be combined in this way. In general, the question of what particle-like phenomena are predicted by a given quantum field theory is a difficult non-perturbative question, and it can change significantly across a phase transition. I don't know what the spectrum of particles would be above the EWSB temperature, and the question about "electric charge" is just one part of this bigger question.
For more detail about the relationship between electric charges and the more-fundamental $SU(2)_L$ and $U(1)_Y$ charges, see this post:
 "Electroweak Phase Transition in the Early Universe?", https://arxiv.org/abs/hep-ph/9611462
 "weak interaction and Parity violation," https://physics.stackexchange.com/a/438967
At the moment the standard model of particle physics is an SU(3)xSU(2)xU(1). This describes the group structure of fixed charge and isotopic spin etc particles. The quantum numbersin the table are fixed.
In the electroweak epoch
In physical cosmology, the electroweak epoch was the period in the evolution of the early universe when the temperature of the universe had fallen enough that the strong force separated from the electroweak interaction, but was high enough for electromagnetism and the weak interaction to remain merged into a single electroweak interaction (above energies of about 246 GeV).
The quantum numbers before symmetry breaking and after do not change so charges of electrons etc are there. What happens with symmetry breaking , is that before breaking the particles have zero masses and after symmetry breaking the gauge bosons acquire mass, and also some of the particles in the table, among them the electron, due to the Higgs mechanism.