If the real component of $e^{j\theta}$ corresponds to voltage or current, what does the imaginary component correspond to? Do we ever use the imaginary parts of these complex quantities?

The oscillatory $V(t)$ and $I(t)$ quantities we deal with are typically solutions to linear differential equations. The physically real quantities are the (mathematically) real parts of the solutions, but general complex solutions to the equations have imaginary parts too; as we discussed, if $C_0 \cos(\omega t + \theta)$ is a solution, then so is $j C_0 \sin(\omega t + \theta)$ and so is the sum. You can think of the imaginary part of the complex $e^{j\theta}$ quantity as an orthogonal solution to the DE. Keeping the general solution intact is very helpful for doing calculations; it works for manipulations involving addition and subtraction. You keep the imaginary part around during intermediate steps (and for visualizations, like the phasor spinning) but then convert to physical solutions at the end of the calculation.

In some situations the imaginary part will correspond to something physical, but I think for our case we'll mostly be using real parts for voltages and currents.