Questions about Touch id and Screen replacments

Does starting up the phone with a unrecognized touch id sensor permanently disable touch id?

No, you can boot the device without a home button and when you re-install the HB, the phone will recognize it.

Can a bad display cause Touch Id not to work?

Yes. The Home Button connects to a flex cable that runs underneath the LCD heat shield. If that flex is damaged, then the HB will not be recognized. This is especially true when the flex tears at the 90 degree bend just before the connection to the logic board.