Should I screw the wires together before input inside the screw wire nut connector?
No, you don’t need it. Moreover, we do not recommend to do so. Place the wires together and put them inside the connector, screw clockwise. СИЗ-К made by KBT have special wings for easier operation of wires with bigger cross-sections. If you will do the preliminary screw with the pliers, it will do not good. First of all, the wires can be pressed and damaged by the pliers; second, you will not get the ideal screw like inside the screw wire connector. Anyways you have screwed the wires together already you can just put on the insulation tape and consider the connection to be finished, without any need in a screw wire nut connector.