Is the inverse Fourier transform always defined? The Fourier transform of a continuous-time function is defined if the function is absolutely integrable, otherwise it does not exist. What about the inverse Fourier transform? If I make up any function of $\omega$, can I always invert it to some time-domain counterpart? 
Would that mean that every function in the frequency domain will have a corresponding function in the time domain but not vice versa?

Well, you can't just make up any function of $\omega.$ Just as for the Fourier transform itself, the function has to be nice enough, e.g. $L^1$ or $L^2.$ Since the Fourier transform can be extended to tempered distributions the functions can actually grow polynomially at the infinities.
Perhaps you haven't seen it yet, but the inverse Fourier transform is almost the same as the "forward" Fourier transform. They only differ in a sign and perhaps a factor (depending on which definition is used; there are a few variants). Therefore the same rules of what can be transformed apply the inverse transform as to the "forward" transform.
