Thanks for taking the time to post those pics but to be honest they won't help us much. If it was a matter of the bulge it would be really noticeable right away on your part, the issue is if it's more of just a subtle one. Basically just lay it down on a table and then flip it over, and if its more or less just flat on both sides that's what you're looking for. But for the most part any smartphone battery is just going stop holding a charge when it fails, the bulge thing is just a problem that happens every so often but rarely.
Let your phone run in Safe Mode for several minutes and see if it heats up the same way. If not, that's a possible indicator some app you've installed previously has become a problem and is taxing your phone's resources to the point where its heating up.
http://www.samsung.com/uk/support/skp/faq/1043686
After you reboot back into normal mode and take a look at your Settings >> Battery. Do you see anything odd that's running all the time? If you tap on Android System that should show you a listing of various system-related processes. Anything there that might not seem right? I'm still hoping it's a software issue and not hardware related. There's always the last-resort option of a Factory Reset.
Try install this 'Trepn Profiler' app. It lets you run power and performance profiles on your phone. It's a little obtrusive when you're using your phone and you have the these semi-transparent testing boxes on the screen at the same time but maybe something will stand out to help you diagnose what's going on.
https://play.google.com/store/apps/details?id=com.quicinc.trepn&hl=en