in my experience almost always if the prox sensor you took off the old screen isn't damaged, it's the new screen that's causing the issue, try one or two other screens and you'll find one that works. I have no idea what about the screen causes that issue, but I've even taken the screen that the prox wouldn't work with and put it on another phone and it worked just fine.

For board Repair I would suggest iPad Rehab. http://www.mendonipadrehab.com/ They are very good. You have to send them your device but it'll be worth it, I've use them a few times so give them a call.

I would say it's not guaranteed, but does the unit turn on at all? The safer bet would be to buy a thunderbolt cable to plug in to an external display to see if the computer still works first before purchasing a screen.

This could be a bad screen from the factory or the digitizer cable may not be seated properly, I would suggest opening it back up and reseating that cable and if that fails I would try sending the screen back for a replacement to the company you got it from.

First thing I would try is a new battery to eliminate that as a cause. What it most sounds like to me is what Ben mentioned in his comment, about the chips near the battery connector either getting knocked off or damaged . Those chips can be replaced though if necessary, but being that it's an iPhone 5c it could very well be cheaper to replace the phone that have micro soldering done on the board. If you can post a picture of the circuit board near the battery connector, I can let you know if anything has been damaged or knocked off. Let me know if this is helpful.

Have you tried a new charging port? Also I have had this happen in a phone I was repairing before and all it took was blowing the circuit board off with compressed air and everything worked fine. The charging port flex cable includes the port headphone jack and the mic and sometimes that can help.

I would try and pry it out carefully with pointed tweezers : https://www.amazon.com/URBEST-Compact-Straight-Extra-Point-Tweezer/dp/B00NL8VX68/ref=sr_1_4_s_it?s=...-4&keywords=point+tweezers try to keep the tweezers in the center of the port hold when prying the piece out because there are pins on the side wall of the port that can easily be damaged. There are shops that do this type of thing for free. I do at my shop. I would call around and ask a few different shops.

Can you remove the battery? If you can, try and remove the battery and then put it back in, do not press power button. If the phone turns on without you pressing the power button, then it is likely that the power button is bad. If this doesn't resolve the issue, I would try a new battery. If that doesn't solve the issue then see if you can get the phone to boot into the recovery menu and wipe the phone. Start by switching off the cell phone by using the Power key. This should be the instructions for the recovery menu: Then press and hold Volume Up + Power button until you see an Android logo on your screen. Select using Volume buttons option "wipe data/factory reset", and confirm that with Power button. Select "Yes--delete all user data". Let me know if any of this works and if not I will try and help further.

I would definitely try another company's screen for a last ditch effort and make sure the lcd connector is seated fully. Sometimes the connectors on the aftermarket screens aren't made exactly the same and you'll think they're pressed down all the way but they wont be making full contact. Sounds like you've done exactly what any technician at any shop would have done so far, you're on the right track. Let me know if this helps

I would suggest trying to unplug the power button flex from the circuit board and plugging in the charger and see if it will come on then, if that doesn't work try a new battery, charging port, screen and if all else fails plug it into iTunes and put it in DFU mode and do a restore. you will lose all info.