-This relay is for 575V but will it work with 120V? - Spec sheet shows 100-600V, so it should.
-Does it need to be wired to the AC voltage to trigger the relay? Yes, most solid state relays require a load to be connected,
I don’t think this one is any different.
I don’t like the different color wires connected to the relay- particularly black and white, makes me think something is going to go boom when it does switch on, hopefully you know what you’re doing.