Does the inverse-CTFT exist for a dirac delta?
Yes, this is a complex exponential $e^{2 \pi i f_0 t}$, at a frequency determined by the delta's "position" $f_0$ (your input being $\delta(f - f_0)$). Write the integral for the inverse Fourier transform, use the definition of $\delta$ and you'll see it "selects" at this particular frequency the complex exponential being integrated.