iPhone5s data/3G wont work after factory reset

I bought this iphone5s online, it was PRE USED (i think maybe even refurbished?), it has a 6 month warranty and they are refusing to let me return in.

Two problems exist.

The first has lasted for around 4 weeks now. The screen appeared almost un-calibrated.. 'ghost clicking' sporadically clicking the screen and I cannot control this - the phone has been unusable. I have put it into the phone repair store 3 separate occasions spanning over the 4 weeks - they have found no physical fault and advised to wait until the next software update in 2 weeks.

I read online that a factory reset would solve the problem if it was a software issue - so I did it. Very difficult to reset up a phone when the screen doesn't work. (The problem was not fixed)

I eventually got the phone activated and set up again etc. But now I notice that now the 3G / data does not work.

What I mean by this to clarify, i can slide the 3G data on. But when I click into an app it does not work - it says no connectivity. I can see the top screen shows my service provider and the 3G label. But no connectivity - this worked perfectly before.

I can still make calls, use wifi, send texts etc.

I read somewhere about jailbroke iphones? But the person who sold me this never stated it was jailbroke - and he also wont allow me to return the device (with still a week left of its warranty). So I assume if he didnt state it was jailbroken I would be entitled to return?

But besides the return, has anyone any clue what could cause this and how to fix? Reinstallation of drivers maybe?

I have been on the phone to apple technical and he made me reinstall itunes and update software and I asked would the installation of software cut the call. He said yes, and I said but nothing will of been fixed, he said hes following an outline of steps and this is the solution - but it hasnt worked. Still the same problem