I don't have flair but this is an argument I covered in my Philosophy undergraduate course a few months. The first thing I'd like to point out is that moral relativism is actually a form of moral realism - moral facts do exist, it's just that what determines what these facts are depends upon either the individual or the society in which one lives.
I hope you don't mind me avoiding most of your questions because I think the best way to see why this kind of moral relativism doesn't really hold up is to just pretend it as an argument and see why it's weak.
If we were going to present it as a formal argument it might go something like this:
Different societies have different moral beliefs
Therefore there are no objective (i.e. non-relative) moral truths.
But clearly the argument is not a valid one - you can accept the premise without being forced to accept the conclusion. That's because there's a secret premise hidden between the two, a sort of implied one, which might be written as "If there is an objective moral truth, then different cultures/societies will not have different moral beliefs about everything." Most relativists don't like to admit it because it's patently false.
That makes it a formally valid argument, i.e. that if both premises are true, you would have to accept the conclusion. But I think there are good reasons for resisting it, particularly the second premise. All societies care about morality and try hard to find the truth, but the fact that they have not converged by now does not tell us that there is no truth to be found. Convergence might be possible if everyone was rational, but many people can be very emotive about morality and this distorts the discussion. The historic influence of religion and politics and the like might also have served to slow down this search.
But even aside from that, I think there are some good reasons to reject moral relativism. This doesn't mean you need to become an out-and-out moral objectivist - you could look into Error Theory, Divine Command Theory, or simply reject the existence of morality at all, for example.
The main one, for me, is that it actually eliminates our ability to express disagreement with people. So I mean, think about what you're really saying if you're a moral relativist. When I say "FGM is morally wrong", what I'm actually saying is "FGM is morally wrong to me." And when someone replies "FGM is morally acceptable", what they're saying is really "FGM is morally acceptable to me." But both of these can be true at the same time, they're not mutually exclusive - so relativism actually removes our ability to even express moral disagreements with others.