No, they don't need to be physically connected.
I'm sorry, what you're aiming for is almost impossible. It's very difficult even with good screens, but with POS Mac screens it's a bridge too far, I'm afraid.
I don't have instructions for the S4E, but the S5E ones won't be miles different. Have you tried following them? https://www.damiensymonds.net/cal_S5Elite_mac1.html