Do you have an external DC PSU that you can power the phone from? If you do you can measure the current draw of the phone. Also, if you have a Mac, search for a program called "coconut battery" it will help you diagnose whether your battery is in need of replacing. I have also found that more often than not, when you replace a battery on a 5S, you also have to restore the phone - and no back-ups allowed. Those are some things to consider before you just dive into the motherboard
Get a screen you know works, and connect it to your phone. If the screen works, then the cable is your problem. If a known good screen isn't working, then you know you are in need of a board level repair.
I'm thinking that there are a few more things you can try. In settings, general, accessibility - turn off your "noise cancelling" then try a call and see if that makes any difference Check also that holes for all 3 mics, front and back top, and then the bottom are all clear of dust / debris If all of that makes no difference, you may have a problem with the sound control IC
I've got a few of the essential tools on order, and it's a journey I am about to embark on. I know it's going to take a lot of practice, but I'm really looking forward to getting stuck in. I really wish I could get over to NY and take Jessas practical board repair course, but that's not a luxury I can accommodate at the minute - though I would value that course over anything any collage here in the UK could offer
I can do, and have done several 6s and 6s plus glass only replacements using OCA... If anything they are easier than the iPhone 6 to do, because the backlight is supported by metal, making it a whole lot less flexible than the iPhone 6 LCD, which would then easily break
Noise - because your fan is working overtime trying to cool the cpu Shutdown - the cpu got too hot, so the computer turned off, hopefully to avoid any serious damage It's overheating MOST LIKELY due to excess dust build up around the fan / heat sink radiator area. This is preventing the fan from dissipating the heat from the heatsink