Could you install an application called Elixir 2?
There is a section called "running" you can click on the four lines at the top (settings) and select sort by "CPU". Who is listed at the top when this happens?
This program works similar to TOP on Linux where it refreshes every second or so with the top processes and how much CPU they are using at the time. This is realtime data rather than looking at post incident figures. Do this while it is happening and see if you have a rogue/runaway process/service.
So the next time your phone is dropping battery and heating up significantly run this app and see who/what is the culprit. Report back here when you have some data.
I just had this happen again on a VZW S3. I was just looking for a "TOP" application to see who is hogging CPU and or Memory. Found Elixir 2 and it looks like a pretty good app.