We can derive all of classical electrodynamics (Maxwell's equations) simply from the $U(1)$ symmetry of the electron field?
Maxwell's electromagnetism doesn't require an electron field, it can exist independently of electrons in the vacuum, or it can even be coupled to other fields (e.g. charged scalar fields, gravity, etc.). The choice of $U(1)$ selects Maxwell's theory from a more general class of gauge theories called Yang-Mills theories, which are parameterized by Lie groups/algebras.
We can derive all of quantum electrodynamics (Photons) simply by quantizing the electromagnetic four-potential?
Yes, quantum electromagnetism can be obtained from classical electromagnetism (Maxwell's equations) by applying canonical quantization. There are extra complications due to gauge invariance, you'll need the theory of quantization of constrained systems to consistently quantize Maxwell's theory. But you will in the end obtain a quantum theory of noninteracting photons.
QED (Quantum Electrodynamics) is a bit of a misnomer, because apart from Maxwell field / photons, it also includes a completely different matter field which is the Dirac field, with particles called electrons. These two fields / particles interact with each other in a certain way which is dictated by the gauge symmetry. QED is therefore different from quantized electromagnetism, which can be thought of as an approximation to QED that is valid in situations where the electron field is negligible.
Also note that even though in almost all cases quantum theories are obtained from classical theories by canonical quantization, this procedure is heuristic and has no physical meaning. In reality, only the quantum theory exists. The classical theory is simply an approximation. So to really understand what's going on, one needs to postulate the quantum theory, and then prove that the classical theory holds in a certain regime that corresponds to setting $\hbar \rightarrow 0$ in all mathematical expressions.