Duality of the continuous-time Fourier transform - derivation and notation
You've answered your first question yourself: if you use the notation $X(j\omega)$ then you'll need to use $X(jt)$, because you simply replace $\omega$ by $t$.

The sign error in your derivation comes from the fact that you forgot to change the sign of the limits of the integral. After substituting $\omega=-u$ you should get

$$X(t) = -\int_{\infty}^{-\infty} x(-\omega) e^{j\omega t} \mathrm{d}\omega = \int_{-\infty}^{\infty} x(-\omega) e^{j\omega t} \mathrm{d}\omega$$