I've had those exact problems. Extreme heat during usage and charging as well as terrible battery life. Nokia changed the battery. Same phone. We're good now :wink:
Does it still get hot when gaming?
Look at other threads under battery issue in the head line. It's very common. The problem is the app you're using connecting to internet or there're other apps running in the background that 're trying to connect to the internet. When that happen, battery is drained and the phone get hot. It's very annoying. Also some apps can access to internet while being invisible from background running list and you have no control over it.