In the past, women had to have a dowry to get married. There was probably a reason for that (economic or market reason, I guess). It probably wasn't a very good reason, and likely isn't very relevant today.
Probably, yes. But whatever "economic" reasons there are, the cultural reasons were probably much more powerful.
Maybe men were more scarce in the past than women. Wars might have accounted for that, but on the other hand, men have longer "expiration date" (when it comes to mating). Also, if men were so much rarer, why didn't polygamy develop?
You could claim that women are weaker, so they were less "useful" in the past (when most labor was manual) and hence less valuable. On the other hand, without a woman, there would be no next generation, so each man needed a woman to ensure his survival in old age.
These are the most obvious justifications I could come up with, and they don't sound very convincing. Let me know if you can think of any more.
This statement is logically inconsistent with reality. If "signalling of commitment" was a real need, why would only one party need to provide this signal? And not just one party (random), but a specific party (predetermined) - the man.
You're right, however, that in the broader context, diamonds are an honest signal of affluence - by buying expensive, but useless stuff (conspicuous consumption) we send a strong signal about how much we can afford and how rich(-er) we are.