Check you battery connection. The clip where you battery is connected may have broken free, we've seen this several times. The phone is reaching for the battery for a base charge to start up 1/2 way through the load and isn't getting it. There are several others issues that can cause this type of boot loop, but only a few where the phone is only responsive when plugged in, so this one is most suspect. "careful with self promotion in "Answers" could be viewed as SPAM"
Logic boards are sometimes farmed by repair shops and sold online. If it comes to requiring a new board eBay is a possibility, but buyer beware. The battery is a likely spot first, as it will be far more cost effective. Do mind the adhesive underneath the battery in the iPhone5c/5s as it can be tricky to deal with. PLASTIC tools only on this one! No self promotion
Reseating the clips is correct. Also make sure the battery is disconnected throughout the repair. We have purposely created this lines in the LCD be adding static electricity to the device during the repair, and waiting for the static to dissipate also works. Reseating the clips is a faster way of grounding out the static, so try that.
This actually sounds like the devices proximity sensor isn't functioning properly. If the screen were "too sensitive" then the issue would also be present when the device is away from your face, such as the reported ghost fingers issue. Chances are the device requires a UV screen over the proximity sensor. Some devices, such as the white iPhone 4, must alway have this or the result will be exactly as you've stated. If you need parts for your fix, I'm sure you can get them here at www.ifixit.com
Firstly I would try connecting with your warranty provider. Unfortunately the only way into the lid to tighten the hinges / replace the hinges with new ones is to "break open the lid" This can be done very conservatively, with no tool marks when done, but requires a little skill.
Firstly, check to see if there is a "piece is stuck inside where you press the sensors down overr the little speaker" YES that could be an issue. Here's the goods on error 4013: http://www.wiki-errors.com/err.php?wiki=4013 If you need a replacement part, you can order on through ifixit.com i'm sure.