Does the port accept mouse input? I was playing the browser version and could only control the cursor with the arrow keys.
No, this port doesn't have mouse support. Later models of the Apple II could support a mouse, but I don't have one to test with.
Adding mouse support also would complicate the music playing code, as often the Mockingboard (sound card) and mouse card are expected to be in the same expansion slot (#4).
In theory you can press J to enable joystick mode and use a joystick for the pointer, however that's not well tested and also I'm not sure if it would work at all with the in-browser emulator