I honesly think mermaids exist. They are legendary aquatic beings with a human upper body and a fish tail lower body.
Hey, I'm not one to judge. They just want to be part of our world, right? But mermaids are the good creatures, right? It's sirens we'd have to worry about.
I don't know what a siren is. Mermaids were first mentioned in ancient Assyria. 
Are they part of the mythology of a lot of ancient cultures?
Yes they are apart of European, African, Asian and Near East cultures.
What other features of an average mermaid should I be looking for besides the tail?
Just basically the body of a women up top. The goddess Atargatis turned herself into a mermaid after accidentally killing her human lover. It was totally out of shame.