Is it possible to add text to a sprite inside of the editor screen, or do I actually need this separate sprite text layer on top of it as a separate item?
You need to add a text object on top of the button sprite, yes. You can add them to a hierarchy, then the text will scale with the button, move with the button etc.
And also, what is preventing this layer 2 from appearing when I click on the button (while obviously hovering over it too)?
That's why I suggested to use a different animation when the button is clicked. Something like this:
Button set animation to "clicked"
Wait 0.1 seconds
Button set animation to "idle"
Then changing animation frame will have no effect.