Nyquist frequency , sampling distance
Your answer to the first question is right, provided that the signals involved are real and lowpass (baseband) nature.

The second question is tricky, assuming ideal infinite length sinusoidals, your intuition is correct in the sense that their Fourier transforms multiplied would yield zero (unless their frequencies were the same), and has zero bandwidth. Also note that the convolution was divergent in the first place, if that's a problem for you.

However, any practical implementation of that algorithm would include windows and therefore result of Frequency domain multiplication will be nonzero and the result would be non bandlimited, hence cannot be sampled without anti-aliasing filtering.