Minimum Sampling frequency for this signal to avoid aliasing
There's several different ways you can go about solving this problem just in the time domain. Use trig identities or Eulers formula to reduce the equation. Both are simple enough and just require some time to work through. Feel free to use WolframAlpha to check your work. I find it sometimes helpful for tedious algebra.

Yes, the Fourier transform of a sinc function is a rectangular function, so you will have what you've described as convolved rectangular functions. This is more work than is necessary as after you've simplified the time-domain expression you will be able to find the maximum frequency, and subsequently the required sampling rate, for that signal.