IPhone 5c won't keep time and Wi-Fi keeps randomly dropping, why is this happening?

I ordered a iPhone 5c from ebay and the seller said that it was fully functional. When I got it I tested it out to see if this was correct and found that the iPhone would not keep the time. For example: I turn the screen off at 8:45 and turn the screen back on 5 minutes later and it will still say 8:45 and then the time will start moving. My point is that the time on the phone freezes while the screen is off. Even when I set the time automatically or set it manually it does this. I tried restoring it to factory settings which did not fix it and I found that it will only keep the time if I leave the screen on or keep the iPhone connected to my laptop which is very inconvenient because I wish to carry this with me when I'm out and about. Also I noticed that the Wi-Fi connection randomly drops a lot which is annoying when using apps. It isn't a router problem, this never happens on my iPad or any other device.so I'm wondering if these two issues are caused by internal problems.

The iPhone is with Sprint and I do not have a Sim Card for it since I do not wish to activate it with a carrier yet. (I want to use it like an iPod and backup phone in case my current one breaks) I contacted the seller and they're offering a full refund but I just want to check if I can fix these problems myself first and consider the options I have before deciding to send it back. In the second half of the picture you can see the time hasn't changed and the Wi-Fi isn't connected. Please help!

The world is ending

That's weird

Mine does the same! Someone please help me!

My iPod 5th generation is doing the exact same thing, On Christmas that will make it that I've had it for a year. But I haven't even used it all year, I left it turned off for about 4months, and dropped it a few times. I'm wondering if it's only going to last s year or if there's something internally that can get fixed.