In our modern understanding, every electron is thought to be a localized excitation of the electron (or Dirac) (spinor) field $\Psi(x^\mu)$, while every photon is considered to be an excitation of the photon (vector) field $A^\nu(x^\mu)$, which is the quantum field-theoretic counterpart of the classical four-potential.
Thus, the answer to your questions are:
All particles of the same type (e.g. photons or electrons) is understood to be 'coming from' one all-permeating quantum field. It should be noted that these fields also give rise to the corresponding anti-particles, so the positron field is the same as the electron field.
The different particle types are truly separated in quantum field theory: Each type is represented by one field, and the fields interact. These interactions are quantified by the Lagrangian (density), which essentially determines everything about the theory. In pure electrodynamics, the quantum field-theoretic Lagrangian density is (using 'mostly minus' sign convention for the metric)
$$\mathcal{L}_{\text{QED}}= \bar\Psi(i\gamma^\mu D_\mu-m)\Psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}
=\bar\Psi(i\gamma^\mu (\partial_\mu+ieA_\mu)-m)\Psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}
$$
where $F_{\mu\nu}\equiv \partial_\mu A_\nu-\partial_\nu A_\mu$ is the electromagnetic field strength tensor. The 'covariant derivative' $D_\mu\equiv \partial_\mu+ie A_\mu$ encodes the interaction between the two fields $A_\mu$ and $\Psi$, and the 'strength' of the interaction is given by $e$, the charge of the electron.
The following is not well-known, but (modified) Maxwell equations can indeed describe both electromagnetic field and electrons.
@Quantumwhisp commented: "Maxwell's equations don't describe charged particles at all", and then asked: "Can you derive the Lorentz-Force from maxwell's equations?"
I am not saying these comments are unreasonable, but, surprisingly, Dirac did derive the Lorentz force from Maxwell equations (Proc. Roy. Soc. London A 209, 291 (1951)).
I summarized Dirac's derivation elsewhere as follows.
Dirac considers the following conditions of stationary action for the free electromagnetic field Lagrangian subject to the constraint $A_\mu A^\mu=k^2$:
\begin{equation}\label{eq:pr1}
\Box A_\mu-A^\nu_{,\nu\mu}=\lambda A_\mu,
\end{equation}
where $A^\mu$ is the potential of the electromagnetic field, and $\lambda$ is a Lagrange multiplier. The constraint represents a nonlinear gauge condition. One can assume that the conserved current in the right-hand side of the equation is created by particles of mass $m$, charge $e$, and momentum (not generalized momentum!) $p^\mu=\zeta A^\mu$, where $\zeta$ is a constant. If these particles move in accordance with the Lorentz equations
\begin{equation}\label{eq:pr2}
\frac{dp^\mu}{d\tau}=\frac{e}{m}F^{\mu\nu}p_\nu,
\end{equation}
where $F^{\mu\nu}=A^{\nu,\mu}-A^{\mu,\nu}$ is the electromagnetic field, and $\tau$ is the proper time of the particle ($(d\tau)^2=dx^\mu dx_\mu$), then
\begin{equation}\label{eq:pr3}
\frac{dp^\mu}{d\tau}=p^{\mu,\nu}\frac{dx_\nu}{d\tau}=\frac{1}{m}p_\nu p^{\mu,\nu}=\frac{\zeta^2}{m}A_\nu A^{\mu,\nu}.
\end{equation}
Due to the constraint, $A_\nu A^{\nu,\mu}=0$, so
\begin{equation}\label{eq:pr4}
A_\nu A^{\mu,\nu}=-A_\nu F^{\mu\nu}=-\frac{1}{\zeta}F^{\mu\nu}p_\nu.
\end{equation}
Therefore, the last three equations are consistent if $\zeta=-e$, and then $p_\mu p^\mu=m^2$ implies $k^2=\frac{m^2}{e^2}$ (so far the discussion is limited to the case $-e A^0=p^0>0$).
Thus, the first equation with the gauge condition
\begin{equation}\label{eq:pr5}
A_\mu A^\mu=\frac{m^2}{e^2}
\end{equation}
describes both independent dynamics of electromagnetic field and consistent motion of charged particles in accordance with the Lorentz equations. The words "independent dynamics" mean the following: if values of the spatial components $A^i$ of the potential ($i=1,2,3$) and their first derivatives with respect to $x^0$, $\dot{A}^i$, are known in the entire space at some moment in time ($x^0=const$), then $A^0$, $\dot{A}^0$ may be eliminated using the gauge condition, $\lambda$ may be eliminated using the first equation for $\mu=0$ (the equation does not contain second derivatives with respect to $x^0$ for $\mu=0$), and the second derivatives with respect to $x^0$, $\ddot{A}^i$, may be determined from the first equation for $\mu=1,2,3$.
However, the above is about classical electrodynamics. What about quantum theory? It turns out that modified Maxwell equations can be equivalent to the Klein-Gordon-Maxwell electrodynamics or (with some caveats) to the Dirac-Maxwell electrodynamics (see my article Eur. Phys. J. C (2013) 73:2371 at https://link.springer.com/content/pdf/10.1140/epjc/s10052-013-2371-4 ).
Best Answer
Because photons do not interact, to very good approximation for frequencies lower than $m_e c^2 / h$ ($m_e$ = electron mass), the theory for one photon corresponds pretty well to the theory for an infinite number of them, modulo Bose-Einstein symmetry concerns. This is similar to most of the statistical theory of ideal gases being derivable from looking at the behavior of a single gas particle in kinetic theory.
Put another way, the single photon behavior $\leftrightarrow$ Maxwell's equations correspondence only holds if you look at the Fourier transform version of Maxwell's equations. The real space-time version of Maxwell's equations would require looking at a superposition of an infinite number of photons — one way to describe the taking an inverse Fourier transform.
If you want to think of it in terms of Feynman diagrams, classical electromagnetism is described by a subset of the tree-level diagrams, while quantum field theory requires both tree level and diagrams that have closed loops in them. It is the fact that the lowest mass particle photons can produce a closed loop by interacting with, the electron, that keeps photons from scattering off of each other.
In sum: they're both incorrect for not including frequency cutoff concerns (pair production), and they're both right if you take the high frequency cutoff as a given, depending on how you look at things.