Just guessing, but since Ursula is a witch maybe she had the power to breathe under water, even as a human. As for the second question, I read somewhere that mermaids aren´t supposed to have gills, but some sort of organ in their chest that can operate as a lung, with the ability to extract oxygen from the water. This allows merpeople to 'breathe' water. The article I found even says that "It has been seen in more than one episode, Ariel taking a deep inhale of 'water' because she is out of breath." Though I´m not sure I agree with that last part, because in the movie, when she sings to Eric, she was out of the water for a long time.
link