When replacing a iPhone 6 screen it is super common for the proximity flex cable to have a small tear. I would recommend looking over the cable to see if there are any tears. 99% of the time it is a small tear in the corner closest to the connector. However, it could also be a bad speaker, but i have only seen that with water damaged phones.
Its highly likely that you broke the LCD during the drop. I would recommend replacing the display unit. The part on eBay is about $38-$45USD. If you do the repair yourself just take your time and don't force anything too much.
Check the home button flex that plugs into the motherboard that could be loose or something. If that doesn't change the issue I recommend getting the back LCD plate replaced. That could be the issue. Normally, I see this problem on iPhone 6s because the home button flex is within the display unit. Let me know if this fixes the problem!
First, open the phone up and clean any corrosion you see using 90%+ alcohol. Rice dries the phone out a little, but not completely. Second, get a replacement screen. It is very common after water damage for the LCD to be damaged. I recommend powering off the device until you open it up to make sure all water is out of it and the corrosion is gone.
Yes the screen could be faulty for the random touching, no touching, and freezing. However, for the auto brightness that is dealing with the proximity sensor. Make sure the sensors are seated well into the screen. and look for any damage to the proximity sensor flex cable. If there is damaged you can get a replacement for $7-$15USD on Amazon. But the screen is a bad screen you need to return it and get a different one.