Yes, both sentences are correct but each has a different meaning. The first is referring to mermaids in general. It is not referring to a particular or specific mermaid. It is simply talking about the existence or not of any mermaid anywhere. The 2nd sentence is more specific. It is not talking about any mermaid or mermaids in general. It is talking about some specific mermaid(s) that was being discussed earlier in the conversation or text. For example, people may claim to have seen mermaids on a particular island. Here, I can say "I do not believe the reports of the mermaids' existence.". "Mermaids' existence" here is referring to a particular or specific group of mermaids. I am not talking about the existence in general of mermaids (perhaps they do live elsewhere). When I say *mermaid's existence*, I am referring to these particular mermaids.

I was under the impression that the genitive ('s) is used only when we are referring to people (e.g. That is Natalie's house) And when we are referring to a thing you should use the preposition "of". (E.g. The wheels of the bike.) So I'd say that the answer would be defined by the context you are giving to the mermaids (if you consider them as humans beings you could use the genitive or if you are considering them as not human beings you could use the preposition of).

I would go with #1 but I would delete "declaration" and change "proof" to "evidence" to improve flow.