HP lists a lot of suitable devices on https://www.hp.com/ch-en/shop/product.aspx?id=3PN48AA&opt=UUZ&sel=ACC, but not a single Spectre among them. I myself use the adapter with a Spectre 13" without any problems. The USB-C connection is certainly important. To be on the safe side, it is better to ask HP.
No the power cable is not included in the scope of delivery. This would be the matching cable Lindy Notebook power cable (3-pin type J, cloverleaf, 2m, connection cable)