Treatment of electrons and phonons in condensed matter physics

You can do the do the whole derivation using quantum mechanics (see for example J. M. Ziman's Electrons and Phonons section 1.3. His Principles of the Theory of Solids probably does the same thing.), and you get the same result. In fact, the derivation looks basically the same.

When you treat the atoms as though they're connected by springs, this simply means that they have a potential energy $\propto x^2$, where $x$ is an atom's displacement from its equilibrium position. We already know what an $x^2$ potential does in quantum mechanics, so the "classical" treatment is simply manipulating the potential energy so that it's in a nice form and then quoting the quantum mechanics result we already know for a simple Harmonic oscillator.

At a higher level, I'd say that all the derivation was doing was finding the classical Hamiltonian of the system and then using that as an input to the Schrodinger equation. That's how all of basic quantum mechanics works: take the classical Hamiltonian of the system and replace the classical momentum with the quantum momentum operator. (The lecturer may have used Newton's laws instead of writing down a classical Hamiltonian, but the two are equivalent.)

WRT the order of the time derivative, I'd say that they're different because electrons have mass while phonons do not. That's true in both classical and quantum mechanics.