For question 1, is omega_c > 0 the same as "there are no conditions set"?

no, because it excludes $\omega_c=0$. The carrier frequency $\omega_c=0$ could be equal to zero, in which case, the signal would simply be multiplied by one. Of course, one can then recover the signal! -pm

For question 1b), should $e^{-j(\omega_c+\theta_c)}$ be $e^{-j(\omega_ct+\theta_c)}$ ?

Yes, you are right. The correction has been made. -pm

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