In this basic classical treatment, you will always get Raman scattering, whether you send in a modulated or DC field. But in reality, all of the details come from the innocuous-looking "oscillator strength", $\alpha_1$. This is the coupling constant, which depends strongly on the details of the problem, including the excitation frequency and polarization, and the crystal/molecular structure and properties.
So real life, you would not get Raman scattering if you excite with a DC field because $\alpha_1(\omega)$ is a function of frequency,
and $\alpha(0)=0$, as your intuition probably tells you. It so happens that when you do a little more detailed calculation, you find that $\alpha_1(\omega)\propto\omega^4$.
Regarding your second question, if the Raman interaction is based on the electric field of your incident light wave, as most of them are, then any sinusoidally varying electric field of the same frequency (and polarization and amplitude) would give the same result. Once again, this comes down to the details of $\alpha_1$ and whether it is an electric dipole-type interaction for your particular material/molecule. As I said, most of them are, but one can think of materials (like antiferromagnetic NiO) which have a magnetic response instead.