"Faulty home button, doesn't work"に関する回答Error 53 is a vestige of history. It is no longer a problem since Apple released iOS 9.2.1. As long as you're not running iOS version 9.2.0, you shouldn't see error 53. See https://techcrunch.com/2016/02/18/apple-apologizes-and-updates-ios-to-restore-iphones-disabled-by-error-53/
"Screen only works when phone is open"に関する回答My first guess is that a cable on your new display is unreliable: As the cable flexes when you close up your iPhone, one or more conductors in the cable are losing their connection. (1) If evidence shows that there is an iffy conductor, I'd remove and reattach the display, in hope that it's simply a poorly seated connector. (2) To determine if it the simple case that the display is simply not displaying anything, I'd try using a USB cable and see if the device shows up in iTunes even with a non-functional display. (3) If the old working-but-cracked display is still available and viable, I would temporarily re-install it and see if the old screen's behavior is normal.