iPhone 5 Headaches...

So this actually started about a week before getting my iPhone 5. My 4s was running extremely hot, even when not in use, and I saw my battery life go from ~8-9 hours down to like 3 or 4 hours.

When I restored my iPhone 5 from that phones iCloud backup, I experienced the same behavior! I've since given the 4S to my mom (restored) and she hasn't had any problems with it.

I just tried factory restoring, first annoying thing is my texts are gone... but fine. The really annoying thing is in the App store none of my previous purchases are showing up, and next to apps I bought (when I search for them) it has the $ amount and wants me to buy it.

Has anyone had either of these problems? I've tried googling issues or finding stuff on apples forum, but not much luck.