What forms do if file by mail with IL-1120-ST. TurboTax also printed out Schedule B (I assume this is required), but also Schedule K-1-P, and the complete Fed 1120S

Mail all forms and schedules that printed including the federal Form 1120S.  Many state returns require a copy of the federal return as well.

From the New Jersey Instructions for mailing: New Jersey 2017 CBT-100S

(5) A copy of the Federal Form 1120S, including Schedule K, must be submitted with the CBT-100S