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You already asked a similar question before
http://talk.sonymobile.com/t5/Xperia-Z1-Compact/problem-in-heat/m-p/991899#M15997
The longer you use the phone the hotter it will get and the camera wont' work, there's not way around this.
Are you using the phone while charging? playing games? perhaps you have apps that are constantly running, keeping the phone hot.
Simple, boot on safe mode for few hours
http://talk.sonymobile.com/t5/FAQ/How-to-boot-your-phone-into-safe-mode/m-p/348008#U348008
check if the battery drains as fast, also see if it takes as long to charge the phone, another thing to check, do you keep wifi on aat all time?
Boot on safe mode for few hours and reply back, WiFi will drain your battery faster
Settings > power management > battery > check the graph and also the top 5 items on the list, how long have they be running? (hours, minutes) post the screenshots here.
I suggest that you try to repair the phone software using PC Companion..
Before repairing your device you may want to backup your information first. Check out this topic for more information on how to.
How to backup?
If the issue should still remain I think that this needs to be examined and fixed at a repair center. For more information about how to submit your phone for repair and where, contact your local support team.