I have replaced and repaired near 40 iPhone 5c and have never had a problem with fixing them would rather have them than a iPhone 5 anyday(repair wise)..
Neguy when a iPhone chirps when pluged into a charger that means it needs a new display there's nothing wrong with the phone just needs a screen. When you popped the screen back in you must have damaged the ribbon cables..
Ok soya now I would say a new battery will sort you out it's not that expensive to find out..
A battery can get damaged when the phone gets smashed it might not show it straight away sometimes but can degrade and sometimes like you have experienced just shut off..
You could try apple for warrenty but because the phone is smashed it might void this but try your luck hope I have helped a little bit