I have seen a similar issue with other touchscreen devices. Most manufacturers offer a calibration program that is supposed to fix this issue (supposed to but doesn't always for me). Make sure the screen is cleaned properly since grime and dust can sometimes mess with the capacitive touch detection. Sometimes if you turn it off and clean it then use gloves or a thin cloth to configure the touch calibration it is less sensitive to atmospheric changes and statically charged dust. This can also be caused by a screen fault that necessitates the replacement of the screen (or you can just turn off the touch part of the display). I have sent so many devices back to the manufacture with this "Ghost" issue and it's never consistently clear what causes it.
Hi Ellamama Replacing the screen and digitizer is a solid idea; the part you specified also comes with the small forward facing components (camera, mic, speaker etc) . I repaired a phone that was dunked and the screen stopped working after the replacement the screen worked but siri didn't work and talking on the phone was intermittent because the front mic and speaker were damaged and did not get replaced. As far as the battery goes the water damage looks like it's on the opposite side from the battery connector so it probably did not reach the power circuitry. You could test it in another phone or with a meter. Your picture shows some clear residue from whatever was spilled on the phone... I think you should be able to determine how much of the phone is affected and if it needs cleaning. Personally if she has been using it in this state whatever is going to be damaged has been already and since it still works (mostly) nothing on the main logic boards has been compromised. If the phone is shutting off...
When you opened the phone there should be a small ribbon cable that went from the screen assembly to the logic board that connects the home button (and the touch ID) to the logic board. Unplug and re-seat this then make sure you restart the phone (unplugging the battery and re-plugging it might also be a good idea) this should force your phone to reload the hardware and detect the home button again. I have done this several times to older versions of the iPhone. I think simply opening the case can sometimes loosen the cable enough to cause the home button to stop working.
Hi R P Its sounds like you might have damaged the power circuitry. Does the power LED light detect that the battery has a charge when it is not plugged into a power source? Does the computer sleep on battery power or turn off completely when you unplug it? If you did damage the power circuit you can run a hardware test to see if it detects any problems. https://support.apple.com/en-us/HT201257 The main power circuit is integrated into the motherboard so keep in mind there is not a lot of options for replacing it.
It is soldered directly onto the logic board. You can solder a new one on or replace the whole board but other then that your options are pretty limited.
You might want to try unplugging the battery and holding down the power button for a bit (like 30 sec or so) to try and reset the hardware so when you plug it back in it will re-detect the hardware in the phone including the battery... I have had this work on other phones but I am not sure about the Nexus. Good luck!