Seeing the actual screen that displays your picture is damage you have to replace it all, how most companies do it is they glue the part that displays what's on your phone (The LCD or OLED screen) and the glass together. So it’s all 1 unit. You can find these replacement screens on iFixit’s store or eBay :)
What I’d do is try to carefully open it without damaging anything then disconnect the battery, then after that when you have free time go to a local repair shop to see if they will try a spare battery and screen on it to see if that fixes it. Try doing this before worrying about board level issues, that can make you quite worried about your data. But yeah try a new battery and screen first. If it works then great! :)
I’ve had this very issue with the 5S. Sometimes the screen will go bad and when it does it can cause the phone to appear dead, sometimes it won’t even appear in iTunes, won’t vibrate… nothing. Replacing the screen will 9/10 fix your issue :)
I’d try cleaning it out carefully, try not to damage the flex cable etc. If that doesn’t work see if you can go to a microsolderer or something along those lines to see if they can fix it, I’d personally rather pay or spend time trying to fix it rather than replace it and loose Touch ID. :)
Yup, as the guy above said it’s mainly your WiFi antenna, that is if your talking about the back of the phone at the top. If that’s the case I’d replace it and make sure it’s all sealed up before any dust, dirt etc get inside. It keeps the phone healthy to be clean inside :)
If you don’t know the lock screen password you’ll have to find a computer to reset it through. Once that’s done enter his iCloud info, which the owner should know, if they don’t Apple requires you to have multiple security options to reset the password. So in short term, reset and put in the info and then your done. If you don’t know the iCloud info and for some reason can’t recover it. It will be completely useless….. forever… that is, unless you have the original receipt, in that case you can call up Apple, provide proof of purchase and then your done. :)
So in short term, seeing you replaced the battery, I’d say if you used around 3 strips evenly spread along the battery it should be fine and stay held in, if you used less or prefer adding more to make sure it stays there feel free :)
So in short term the actual screen that displays the picture and the glass are glued together, unless you have an expensive machine or amazing patience and the skill of not breaking the screen (which is very easy to break) you will need to buy a new screen. If you think the iFixit kit is expensive there are plenty of cheaper ones on eBay, but you might not get the same quality, if you do get that route either make sure it comes with the tools needed or you buy them separately. A YouTube tutorial or the iFixit one should help you out. Hope you get it fixed :)