How do sufficiently energetic photons form electrons-positron pairs? If the question seems strange, here's the wikipedia snippet that drives it:

The Big Bang theory is the most widely accepted scientific theory to explain the early stages in the evolution of the Universe.[133] For the first millisecond of the Big Bang, the temperatures were over 10 billion Kelvin and photons had mean energies over a million electronvolts. These photons were sufficiently energetic that they could react with each other to form pairs of electrons and positrons. Likewise, positron-electron pairs annihilated each other and emitted energetic photons:
γ + γ ↔ e+ + e−
An equilibrium between electrons, positrons and photons was maintained during this phase of the evolution of the Universe. After 15 seconds had passed, however, the temperature of the universe dropped below the threshold where electron-positron formation could occur. Most of the surviving electrons and positrons annihilated each other, releasing gamma radiation that briefly reheated the universe.[134]

(where citation 134 is Silk, J. (2000). The Big Bang: The Creation and Evolution of the Universe (3rd ed.).)
Is this how electrons were originally formed? Is it a fundamental process where, if you get two photons hot enough and smash them together, you produce an electron and a positron?
 A: I don't know, if someone took your question title literally,  that they could tell you exactly "how" the above reaction  occurred. How particle conversion happens  is not understood, but the rules behind it are known. On Earth, we are in the low energy regime, so we have to predict and explore higher energies that we normally would not "see", so the rules may be , and probably are, different there.
Provided the conservation rules of charge are complied with, in other words, if you start the process with a neutral overall charge, and end it with an overall neutral charge, then it's possible, given enough energy,
(or hot enough), as you put it.
From Pair Production

Pair production is the creation of an elementary particle and its antiparticle, for example creating an electron and positron, a muon and antimuon, or a proton and antiproton. Pair production often refers specifically to a photon creating an electron-positron pair near a nucleus but can more generally refer to any neutral boson creating a particle-antiparticle pair. In order for pair production to occur, the incoming energy of the interaction must be above a threshold in order to create the pair – at least the total rest mass energy of the two particles – and that the situation allows both energy and momentum to be conserved. However, all other conserved quantum numbers (angular momentum, electric charge, lepton number) of the produced particles must sum to zero – thus the created particles shall have opposite values of each other. For instance, if one particle has electric charge of +1 the other must have electric charge of −1, or if one particle has strangeness of +1 then another one must have strangeness of −1. The probability of pair production in photon-matter interactions increases with photon energy and also increases approximately as the square of atomic number of the nearby atom.

From Matter Conservation

Because of momentum conservation laws, the creation of a pair of fermions (matter particles) out of a single photon cannot occur. However, matter creation is allowed by these laws when in the presence of another particle (another boson, or even a fermion) which can share the primary photon's momentum. Thus, matter can be created out of two photons.
The law of conservation of energy sets a minimum photon energy required for the creation of a pair of fermions: this threshold energy must be greater than the total rest energy of the fermions created. To create an electron-positron pair, the total energy of the photons must be at least $2m_ec^2 = 2 × 0.511 MeV = 1.022 MeV$ ($m_e$ is the mass of one electron and c is the speed of light in vacuum), an energy value that corresponds to soft gamma ray photons. The creation of a much more massive pair, like a proton and antiproton, requires photons with energy of more than $1.88$ GeV (hard gamma ray photons).
The first published calculations of the rate of $e+–e−$ pair production in photon-photon collisions were done by Lev Landau in 1934. It was predicted that the process of e+–e− pair creation (via collisions of photons) dominates in collision of ultra-relativistic charged particles—because those photons are radiated in narrow cones along the direction of motion of the original particle, greatly increasing photon flux.

In general, if there is not a law that specially bans it, then any process can theoretically occur, although exotic particles will decay very quickly, so for example, the Large Hadron Collider was designed to detect the by products of the decay of the Higgs Bosons, and not the particle itself, which lasted a minute length of time.
A: Pair production is not something where the rule is 'go see what Wikipedia says'. Wikipedia just tries to summarize what is known, sometimes well, sometimes not. 
Pair production is a well understood  process in quantum field theory and quantum electrodynamics, and actually for particles other than electron-positron pairs, also from other parts of the standard model of quantum physics. There is nothing mysterious (anymore) about pair creation, pair annihilation or other interactions between charged particles and photons. 
Quantum electrodynamics (QED) developed by Feynmann, Schwinger and others is the quantum field theory of the elctromagnetic interactions. In it you can calculate the cross sections, related to the probabilities of some creation, destruction and/or scattering event happening between charged particles (electrons and muons, and their antiparticles, plus other particles with internal structure when the internal structure effects can be approximated or ignored) and photons, the carrier of the electromagnetic force. It's a complex set of calculations using quantum field theory (QFT). Feynmann introduced his Feynmann diagrams as a way to make the setting up of the calculations easier. One example of a Feynmann diagram can be seen in the wiki article on Pair Production, at
https://en.m.wikipedia.org/wiki/Pair_production
One of the basic concepts in QFT, QED, and really all of relativistic quantum theory, is that particles (which are really excitations of quantum fields) can be created if there is enough energy for them to arise. Thus, it is really a many body problem, with all the different variations on which particles can be created and how likely those are. It is amazing that QFT and QED, a part of QFT, could figure it out. But that is the difference between non-relativistic and relativistic quantum theory, particles can appear and disappear. 
Those cross section or probabilities, for instance of two photons creating an electron positron pair depends on the total photon energy in their, center of momentum frame of reference, and other factors. They've been demonstrated and measured in the laboratory. The same is true for the reverse process can also be calculated. But the difference is that an electron positron pair always has enough energy to create photons from their rest mass. Photons can only create massive particle pairs if they have the minimum energy to create two massive particles - for the electron positron pair it's 1.022 MeVs total. 
Not every process you can think of can happen. Feynmann diagrams are a good methodology to see what processes can happen, and calculate their probabilities. Typically it is processes where, for QED, charged particles and or photons are involved, and energy, momentum and angular momentum can be conserved. It can be more complex for weak and stron interactions where you have to take into account the conservation of various quantum numbers also (such as flavor, in the strong interactions parity, etc). 
At a high enough energies those processes of photon pair production will happen in large quantities, and they did in the early universe. But after it expanded and cooled some, their energies became lower and with many photons not having the minimum required energies pairs were not created. Pair annihilation kept happening until most positron disappeared. There was an abundance of electrons over positrons initially (why is a whole other story, it is not totally clear yet why there were and are much more particles than antiparticles. It may be to the broken parity and charge symmetry in the weak forces, but it is a work in progress)
The wiki article has more that that, it also has a brief summary of some of the calculations, and conservation laws, used, and what some of it means. QED was a huge accomplishment in physics, and it won its discoverers (or developers, it's just a word) Nobel prizes. It is something we know and should be proud of, there's still plenty we don't know. 
More on QED at another wiki article at https://en.m.wikipedia.org/wiki/Quantum_electrodynamics
And a more detailed review of electrons and positrons in physics and astrophysics at http://www.icranet.org/misc/Scientific_Report_2009/Reports/08Xue_a.pdf
To really understand, and if you wish, to use, QED it is best to go through one of the Quantum Field Theory books. Wikipedia is just a quick view at thingS, and it may not always be right. 
