In Melody mode, the following buttons affect directly the chord table:

advance / rewind the current chord

go to the top/bottom of the chord table

copy the keyboard to the current chord.

delete the current chord

insert the keyboard state as a chord in the chord table immediately before or after the current chord.

Unlike what happens in Chords mode when clicking on a chord in the chord table which causes the program to switch to Table mode, in Melody mode touching a chord in the chord table updates the keyboard and staff to reflect the chord in the chord table.

In Melody Mode the down arrow plays the song in the chord table one chord at a time.  It is used not just to play the melody, but to coordinate with the chord mode when adding chords to melodies.  See Entering chords in The Best Way to Enter Full Songs.