Quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics that is the first theory where full agreement between quantum mechanics, special relativity and electrodynamics is achieved.
$$F=\frac {q^2}{4\pi \epsilon_o r^2}$$
At long distances and low energies the equation you give for the electrostatic force remains unchanged. At short distances and high energies the electrostatic force gets stronger, however there is no simple equation to describe how it varies.
The change in strength of the force is usually described by a change in the coupling constant known as running. This will be described in any book on QED. A quick Google found this popular science level description, and I'm sure a more determined Google would find many similar articles.
My advice is that you wouldn't take what the wikipedia says too seriously. In the first place, QED is a relativistic quantum field theory (QFT) and, as stated in standard textbooks --e.g. Mandl & Shaw, Landau & Lifshitz--, quantum field theory (QFT) and quantum mechanics are two different theories. In the second place, there is not "full agreement between quantum mechanics and special relativity" --a point already known by the father of relativistic QM and of QED: Paul Dirac-- [*]
Regarding your questions, the Hamiltonian equation of motion
$$\frac{d\mathbf{p}}{dt} = \mathbf{F}_\mathrm{E}$$
where $\mathbf{F}_\mathrm{E}$ is the electrostatic classical force given by $\mathbf{F}_\mathrm{E}={e^2}/{4\pi \epsilon_0 \mathbf{r}^2}$. This equation is generalized in quantum mechanics to
$$\frac{d\hat{\mathbf{p}}}{dt} = \hat{\mathbf{F}}_\mathrm{E}$$
where the hats denote quantum mechanical operators. The above is one of the Heisenberg equations, which are the quantum analogue of the Hamilton equations of classical mechanics. The electrostatic quantum mechanical force operator is given by $\hat{\mathbf{F}}_\mathrm{E}={e^2}/{4\pi \epsilon_0 \hat{\mathbf{r}}^2}$. The explicit expression for the quantum mechanical position operator depends on the representation used.
The relativistic classical equation given by classical electrodynamics is
$$\frac{d}{dt}\left( \mathbf{p} -e\mathbf{A} \right) = \mathbf{F}_\mathrm{L}$$
where $\mathbf{A}$ is the vector potential and $\mathbf{F}_\mathrm{L}$ is the Lorentz force given by $\mathbf{F}_\mathrm{L} = e(\mathbf{E} + \mathbf{v} \times \mathbf{B})$ for given electric $\mathbf{E}$ and magnetic $\mathbf{B}$ fields.
The corresponding relativistic quantum equation proposed by QED is
$$\frac{d}{dt}\left( \hat{\mathbf{p}} - e \hat{\mathbf{A}} \right) = \hat{\mathbf{F}}_\mathrm{L} = e(\hat{\mathbf{E}} + \hat{\mathbf{v}} \times \hat{\mathbf{B}}) \;\;\;\;\;\;\;\;\;\;\;\;\; (1)$$
with $\hat{\mathbf{v}} = c \alpha$ with $\mathbf{\alpha}$ being the vector of Dirac matrices [#]. Here $\hat{\mathbf{F}}_\mathrm{L}$ is the quantum Lorentz force proposed by QED.
You can find the equation (1) in the Eleventh Lecture in Feynman's QED textbook but with a slightly different notation --Feynman uses c=1 units, do not use hats to denote operators, and uses the dot notation for the time derivatives e.g.; $\dot{\mathbf{p}}=d\mathbf{p}/dt$--.
As Feynman correctly notices the QED equation (1) is "not completely acceptable", because there are some mathematical and physical issues with the QED expression for the quantum Lorentz force $\hat{\mathbf{F}}_\mathrm{L}$.
[*] You would also check the sections 7 and 8 --specially section 8.3 "Does QFT solve the problems of relativistic QM?"-- of this recent Foundations of Physics paper. I completely agree with his conclusion:
Thus, instead of saying that QFT solves the problems of relativistic QM, it is more honest to say that it merely sweeps them under the carpet.
[#] The MathJax engine used here does not render correctly bold Greek letters, but in Feynman's textbook you will find a beautiful alpha symbol in bold font.
It is not only the "electrostatic force" that is developed in QED, but also the other notions. For example, the number of particles is not conserved, but the total charge is. Strong electromagnetic fields can create particle-antiparticle pairs so the strong EMFs are "unstable".
As the number of particles is not conserved, one cannot write a meaningful equation of motion for a given charge $q$ - there is no guarantee that it will not suffer a "transfomration" in course of interaction. Instead, one delas with probabilities to observe this or that in experiment.
As well, there is such a thing as identical particles in QM. That prevents us from "following" a given particle evolution and requires taking into account impossibility of distinguishing identical particles.
Thus, not only the electrostatic force is modified, but the "mechanical equations" too.
