iPhone 5 activation and WiFi problems

I bought the iPhone from a guy. He told me that he was on Facebook when the phone shut down and got stuck on the Apple logo. He said he never had any problems with the phone, except this. He brought it to his friend who checked if there's a problem with the battery, but that was fine. (This is what i know, maybe not the full story...)

Wanted to restore it, but i got some errors, i searched it and found out the proximity sensor cable is broken, so i took it off. After that i made the restore and started the phone. Selected language and country, and after that i had to connect it to iTunes(usually it have to show me WiFi networks, am i right?), when i connected it i got the message "There's a problem with your iPhone." and a link about warranty informations. I was a bit scared. After that i put in a SIM card and tried with that and i got "Device unknown" again with warranty informations, i thought there's a big problem. With some trick(taking out the SIM card when activating) i got an option to activate the phone with WiFi. But... there was no WiFi network, i tried to put the phone near router, no success. Now, i think there's something with the WiFi chip. I was using a non-original cable, if that matters.

I got some errors in the system log: "Couldn't find WiFi device.". What do you think, the problem is really with the chip or maybe the antenna cable? I can't understand why i got error messages when trying to activate, it's related to the wifi problem or something else? When i connect the phone to computer the iTools software shows the iPhone is only 8GB, why is this?

If the guy told me the truth, how can this happen to the phone, and how can shut down the phone simply when surfing on the internet?

I would be really happy, if someone could help me. Sorry for my language. Thanks in advance!