“no + singular noun” vs. “not any + singular noun”

I think this is because you have “what” ahead. the question is asking "what kind". In grammar, if a possessive determiner or possessive adjective is put in front of a noun, the articles (a, an, the) shouldn't be used. eg. my book, Tom's chair, etc...

Back to your example "What kind of flower is this?", "what" plays the role as "my or Tom's".

hope this help a bit.

They are both grammatically valid, but you have to be aware of the logical context. They are both correct from a strict grammatical perspective, but the first one makes more sense and sounds much more natural in this particular construction. In the context of a comparison ("is better than") the reader needs to know what is being compared. When you use the plural, it sounds like you're comparing one group to another, and it is not clear what the group is.

Typically there is only one man and one woman who would be in a relationship at any given point in time, so the singular sounds better. When you say "no men are better than women at maintaining relationships", it implies there are multiple men involved in the type of relationship in question. Unless the context specifically relates to same-sex relationships or group relationships only, it's confusing and doesn't seem to fit the logic of the proposition.

There might be instances where comparing groups like that might make sense. For example, you could say "no Americans are better than Canadians at fielding a hockey team". The literal implication is that you could select any group of Americans and any group of Canadians and that would always be true, so probably it's logically false, but because people play hockey in groups it makes a little bit more sense. It's the sort of thing a drunk Canadian might say instead of "Canadians are usually better at hockey than Americans."

Where "no men" makes more sense. Outside of the comparison context, it is easier to find circumstances where "no men" makes sense and would be the more natural usage. For example, "no men are allowed into the women's restroom" sounds good and it is clear that you're referring to all men everywhere.

In general the construction "no man" sounds a bit like a proclamation. It is not unheard of for ordinary use, but perhaps more something a king or philosopher would say. So the phrasing "no man is allowed into the women's restroom" is fine in a technical sense, but sounds a little bit like a sign on a royal bathroom (or an English language learner). The phrasings "no man has set foot on Mars" and "no men have set foot on Mars" are both equally valid and natural. Because going to Mars is a grander undertaking, the more dramatic phrasing of "no man has set foot on Mars" becomes more appropriate than when you are discussing a bathroom.