Electrostatics in less than 3D is a fascinating subject. However, the extension to spaces with dimensionality different from three is not unique.
On the one hand, the most obvious choice is to introduce a smaller dimensionality as the effect of dealing with special 3D charge distributions that require less than three coordinates to be specified. In this first case, the lower dimensionality reduces the degrees of freedom needed to describe a spatial configuration.
This is the case of uniformly charged infinite parallel lines and uniformly charged parallel planes. In the former case, two coordinates are enough to specify the position of one of these lines; in the latter, one is enough to specify the position of a plane. Therefore, the dimensionality of the space configuration space is $2$ and $1$, respectively. All the relations and physics dimensions remain the same as in the $3D$ dimensional case. In particular, it remains meaningful to speak about point-like charges interacting through the usual Coulomb potential. The interaction laws between infinite parallel charged lines and planes could be derived from the usual Coulomb law in the limit of continuous line and surface charge densities. Of course, the Gauss theorem still requires a 3D volume and a 3D surface.
Therefore your formula
$$
E(r)=\frac{1}{2\pi \epsilon_0}\frac{Q}{r}
$$
is not correct if $Q$ represents a 3D point-like charge. It should be written as
$$
E(r)=\frac{1}{2\pi \epsilon_0}\frac{\lambda}{r},
$$
where $\lambda$ is a line charge density, such that its 1D integral gives a result with the same physical dimension as a 3D charge.
On the other hand, one can imagine a different case, not based on the physical 3D charges but much closer to the formal structure of the 3D electrostatics: a model based on the validity of Gauss's law in D dimensions. In such a case, it is not Coulomb's potential in real space that remains valid, but its Fourier transform proportional to $ \frac{1}{k^2}$. Therefore, the equivalent of Coulomb's law becomes
$$
\begin{align}
\phi(r) &=-q_1 q_2\ln r ~~~~~~~~~ {\mathrm{in~~2D} } \\
\phi(r) &=~~q_1 q_2 \left| x \right| ~~~~~~~~~ {\mathrm{in~~1D} }
\end{align}
$$
In such a case, there is no reason to maintain the same physical dimensions for charges as in 3D, and it is impossible. However, this is a more coherent, fully 1D or 2D theory.
Notice that, although real 1D or 2D electric charges are impossible, other physical systems can be well described by such lower-dimensional electrostatics. The simplest example I can think of is the interaction between line vortexes with a logarithmic behavior.
