Random reboots and screen weirdness.
Got the iPhone 5C with a busted-up screen and a completely dead battery. Case isn't cracked but one corner is banged up. Replaced the screen, replaced the battery, and it started up/worked great. The moisture sensors weren't tripped and all of the ports and buttons worked so things were looking up. Then I noticed some weird things.
First thing I noticed was, when the phone was being charged it would occasionally spontaneously reboot. I'd plug it in, unlock it, and set it down on the table. After a random amount of time somewhere in the neighborhood of a few minutes, the screen would go black and the Apple logo would appear. It just rebooted all by itself. In experimenting with this I noticed that if I connect the phone to a more powerful charger (such as a 10w or 12w iPad charger) the reboots went away or at least were seriously reduced. (TBH I didn't stare at the iPhone while it charged the whole time.) And then once — just once, so far — last week the phone just went into a constant reboot cycle over and over again for a couple of hours. Then it was fine.
Second thing I noticed was, sometimes the touch screen doesn't react. This is going to sound weird but it "feels" like CPU lag, like the phone is doing some major CPU processing without enough RAM and touch input is lagging behind. It "seems" to have something to do with screen orientation; if I'm using the phone in portrait mode this touch issue rarely happens, but if I rotate it into landscape mode it happens more often. I replaced the screen again but got the same results so I'm pretty sure it's not a screen issue.
Other than these issues the phone works perfectly. I've done a full factory reset, including putting it into recovery mode and clean-installing current iOS.