2) The central limit theorem applies to the distribution of the sample average. It applies whenever the samples are iid and the second moment is finite. The fact that the samples are coming from a mixture of normals doesn't change that.
"Bimodal" doesn't really have a precise definition or test, if you don't assume normal distrubtions.
That paper argues that only if means are separated by 2σ should the distribution be considered bimodal.
But there are many measures of bimodality. [1] [2] [3] [4] [5] [6]
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In any case, I would be very much surprised if the population couldn't be selected enough (age, race, country, diet, family) to have human height be bimodal by any measure.
[1] https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/97WR...
[2] https://journals.sagepub.com/doi/10.4137/CIN.S2846
[3] https://link.springer.com/article/10.1007%2Fs11207-008-9170-...
[4] https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1469-1809....
[5] https://link.springer.com/article/10.3758%2FBF03205709
[6] https://esj-journals.onlinelibrary.wiley.com/doi/abs/10.1007...
Yes and no. The sample must be independent and identically distributed. In your case the "identical" part is not correct, as men and women have different distributions (both are normal but with different mean and std). However, if both distributions are normal, then their sum is normal (even with different mean and std).
The fact that the sum is normal in this case has nothing to do with the CLT - it's just a quirk of the normal distribution that the sum is normal. Had men/women had non-normal distributions with different means/stds, then the sum would not be normal.