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I'm sorry to hear about your issues, but the phone can't overheat nor melt unless it has been tampered with and/or has been exposed to water and the flaps were NOT closed properly and/or if there was a crack.
As for the Xperia Z3, you dropped the phone, it could crack or not, it doesn't matter if it has a case or a film protector, not just Sony but other brands, just google it.
As for battery life, I have great battery life with my Z3 but if the phone is older than 2 years, the battery will not retain as much charge as it used to, that's normal with any brand.
You can always get your device inspected by your Local support they can tell you if there's something wrong with your device, such as a factory issue, which unless the device is new, it's unlikely that that's the case, it could be due to the drop or an old battery.
This is a user based forum, you don't have to convince me nor you should assume that I go easy on Sony, again, you can get your device inspected by your Local support, that will prove who is at fault, what's there to lose?
If you truly have issues with your battery life, do the following and answer how old is your device
Charge the phone 100% use it as you normally would > once the battery is down to 50% >
settings > battery > tap on the battery icon > you can see what has been using up the battery
take a screenshot of this graph and list > tap on the top 4 items on the list > take screenshots > post it all here
Also, go to settings > about phone > status > signal strength > take a screenshot and post it here.