The mouse is called the mouse because before it was wireless they were all corded and the cord was supposed to be the tail which makes it look a little bit like a mouse or a rat but rat would sound disgusting hence the name mouse
The first Mickey Mouse short premiere was November 18th, 1928. It was called, "Stemboat Willie." This short was remade shortly after though when Walt Disney discovered he could make sound cartoons and "Steamboat Willie" was made with sound.
While there is no official name to the sound and noise that a mouse makes, it is generally called a "squeak" based on the squeaky nature of the tone of the sound. These squeaks help a mouse communicate all different sorts of emotion ranging from happiness to sadness to even hunger.