If your battery is very hot this may damage your battery lifespan! If your screen dropping brightness while your phone is hot, try to immediately cool it as long as your brightness return. Don't play heavy games while charging your battery! Or you can play but only if battery will shown 100%
P.S. Try to find new battery (new) and replace the battery. Battery must be charged from 0% to 100 at least once per month! If not then the battery will loose her capacity! That's why new phones from Apple may have battery issues. In the store shelf iPhone may lying month or years who knows? That's why ours iPhone having issues with their batteries.
After seeing my colleagues iphone battery issues, I never buy iPhone, only iPads of which I bought two. I refuse to upgrade to iOS 7 and my iPads work fine with good battery life. Apple is not interested in improving their batteries. They just want customers to buy a new phone every year or two. Btw I use Samsung phones which allows customer to change to new batteries and Sony xperia with excellent battery life.