Could be many things really...The issue seems to be the overheating which in turn drains the battery.
I personally have great battery life and no heating issues whatsoever even when watching non-stop media. The hot battery could be a result of just having a defective battery which in turn will also drain much quicker. Just remember that your display sucks the life out of your battery when you have it set to higher levels, using the flashlight, heavy web use and watching videos could allow your battery to get too hot though not normally. 4g can draw more than usual from your battery also making it hot but that isn't common. I do know that playing games on your phone will cause your battery to get hot more often as well as constant signal use.
So in a nutshell, my presumptions are(in order):
1. Make sure you are not heavily playing games and talking on the phone all night.
2.Turn off your 4g for a day or 2 (some geographic areas will have more strain on your cell phone battery due to signal resistance).
3.Set your display to medium or lower and check to see if there are any 3rd party apps that are pulling too much battery power.
4.Check your phone charger for frayed wires or any other physical damage. Use a different charging cable AND wall plug for a couple days(remember it always wise to use the OEM power accessories when possible to ensure proper continuity).
Do those steps in order checking your battery performance after each. If after every one of those steps your battery does not improve then I can assure you that your have a defective battery or in rare cases, a defective handset. Luckily for us, our Lumia 820's have removable batteries. Change your battery out and if you still have your problem then it is your phone.
Good luck!
(4 years as a communications electrician in the Navy)