Hey iPhone5help,
I hate to have to say this, but I won't sugarcoat it for you: The DFU restore is the deepest kind of software fix there is. If it doesn't work, there's probably a hardware problem. It could be as simple as a defective battery, or the battery's connection to the logic board may be loose. We used to see that from time to time - a person who had dropped their iPhone, even lightly, who had a problem like the one you're describing, and the battery connector had become ever-so-slightly dislodged from the logic board. We'd open the iPhone, reseat the cable, and the iPhone would be good as new.
I think you're going to have to have your iPhone serviced. I know what the Apple Store keeps telling you, and as an ex-Apple guy myself, I understand where they're coming from - but in this case, I think they're wrong. Then again, I could be wrong - but the whole thing, including the weird screen malfunction, says hardware problem to me.
I hope this at least helps you get on the road to a permanent fix!
All the best,
David