Yes, the stone should be brought up to temp, otherwise you'll have uncooked pizza in the middle, or more likely, a broken stone going from room temp to 600* in a few seconds.
Not sure about the kettle, never used that. Without a diffuser like the plate setter, direct heat might be too much on the stone. Might. I just don't know.
The death of "willkat98"
Bandera, M-BGE, WSM, Gold Kettle, Kenmore Gasser, 36Qt Turkey Fryer, ECB, Dutch Oven