Answer to "Why won't iPhone 5c turn back on after turning off? Restored 3 times"did you try a force start? While holding the Home button, press and hold the Sleep/Wake button, until the apple logo appears in about 15 seconds. If your phone refuses to charge, it is more likely a port replacement which is quick and simple, no micro-soldering, and about $60. If that is not the problem then it is the U2 IC chip, which requires micro soldering, and I don't know the success rate, if even possible.
Answer to "Phone won't charge after plugged to the wall"If it is not the battery or the USB cords, there is a charging port device in the phone which is an inexpensive replacement. Call someone who can service iPhones in your area. Unlikely but more serious--a U2 IC chip is bad--which may have to be replaced by Apple.