How do mermaids breathe?

Since their posterior portions are typically depicted as fish-like, one would presume they eliminate similar to fish, so perhaps some research on fish digestion and elimination is in order. Since they are half fish, and a fish's anus (or it could be a cloaca instead) is about 1/4 the way to their tail, then that should be where they eliminate, and preferably into the sea.

That said, there are no real mermaids. They are all fiction.

The good thing about imaginary creatures is that you don't have to do al the detailed engineering to show how they work. No one cares about the impact of square and cube laws on giants, the wingspan of flying horses, or the creation and loss of mass as Alice of ALice in Wonderland grew and shrank. An example of what happens when the engineers get turned loose on this type of question is shown in Larry Niven's "Man of Steel, Woman of Kleenex."