Do you need headphones?
Not usually. Laptop and phone speakers work, but headphones help in a noisy room and with low notes, which small speakers lose. Avoid Bluetooth for the Audio Reaction Time Test: wireless headphones hold each sound back a moment before playing it, and that delay lands on every round.
Do you need to read music?
No. The Melody Memory Test only asks you to compare tunes by ear, and the Chord Recognition Test puts the chord names on the buttons. The Perfect Pitch Test is the exception: naming the note is the test, so you need to know the note names.
Is this a hearing test?
No. The sounds come from your own speakers or headphones, at whatever volume you chose, and nothing is calibrated, so a score says nothing about the health of your ears. For a real hearing check, see an audiologist or a doctor.