Short answer is sadly No. You would have to take the screen apart and micro solder a new cable on place. All requiring special (and expensive) equipment and knowledge on how to do it. You could probably get a repair shop that does both refurbishing of screens and microslodering to fix it, but it will probably be more expensive than just buying a new screen.
The first thing I would try, is disconnect the battery and leave it for about 30 seks and connect the battery again. If that does not work, I would try with a different charger and make sure it the correct charger for you Mac (not an 45w air or similar)
That will absolutely not fit. That being said, I see there are som china copy housings available for your model with a glass backside on Ebay. I would be skeptical about the quality of those, but if its important to you, you could give it a go.
Usually its either a bad connection from the digitizer (one of the two cables coming from the screen) or the digitizer itself has been damaged/stopped working. I would first try disconnect the screen, clean the connectors and reconnect. If that does not help I would try with a new screen and see if that helps.
You cannot replace the homebutton to regain TouchID functionality. The home button that came with the phone is the only one that works with your motherboard. (1) Remember that metal shield plate with a cable running trough it that you took of the back of your old screen? Did you damage the cable there? Try with a different one if you can (2) Check the ribbon cable and controller on the removed button closely. Is there any tear damage visible? If not, then you might still have damaged the controller. If you have a repair shop that does micro soldering, they might be able to help with the last one.