Do all poppys have some ammount of narcotic in them?

Not all poppies contain morphine alkaloids. The strain you are looking for is Papaver Somniferum, if by chicken poppies you mean the "hens and chicks" strain, then you definitely have an opium poppy, but it doesn't mean they'll necessarily produce more opium nor does it guarantee the content level of morphine alkaloids in that opium. Arguably the strongest strain is the Tasmanian Purple, which has up to 30% (or more?) alkaloid content, and is the strain used for the pharmaceutical opiates many of us so pionately strive for. :)