Derivation of Ohm's Law
Ohm's law $\vec\jmath=\sigma\vec{E}$ can be derived rigorously in the limit of small electric fields using linear response theory. This leads to Kubo's formula for the electric conductivity, which relates $\sigma$ to the zero frequency limit of the retarded current-current correlation function.
$$ \sigma^{\alpha\beta}(q)=\lim_{\omega\to0}\frac{1}{-i\omega}\left\{\frac{ne^2}{m}\delta^{\alpha\beta} - i\langle[j^\alpha(\omega,q),j^\beta(-\omega,-q)]\rangle \right\} $$
(This derivation, of course, involves more than just Maxwell's equation. This is properly derived in the context of non-equilibrium field theory.) The Drude model is a model for the spectral function of the current-current correlation function in terms of a single ``collision time''. This model can be derived within kinetic theory, which is applicable when interactions are weak and the correlation function can be computed in terms of quasi-particles.
No, not in the way you are probably thinking. You can do a lot with Maxwell's equations, but you have to step outside of them to derive Ohm's law. There is a trivial relationship going from points to macroscopic objects (e.g., multiplying by lengths and cross sectional areas), but this is just giving different forms of what is still referred to as Ohm's law.
As I pointed out in a comment on Thomas' currently-accepted answer, I think that Kubo's solution implicitly assumes (i.e., does not derive from scratch) a linear relationship between the current and field. It is already is going way beyond Maxwell's laws.
A full answer requires going even beyond that. See, e.g., Riess (2004). So that's why I'm saying no is correct the answer to your actual question.
Importantly, I don't think Kubo's original paper on this attempts to compute any actual values of $\sigma.$ So, neither aspect of Ohm's law was really derived by Kubo. Rather, Kubo's formalism allows computation of $\sigma$ assuming a linear relationship should exist.
For these reasons, I would object to Thomas' use of the phrase "derived rigorously" in describing Kubo's contribution as described. This is also partly why I think my own answer is worth submitting. (I am somewhat bothered by the use of the phrase in this context, especially if also saying that the problematic Drude model also gives it, like it is a trivial equation to derive or something.)
No, it is an approximation and not derived from first principles. It is based on empirical observations.