Only if you root, 'killed' your device, and they know it came from rooting, then maybe they have a list which devices were broken by user end
For the heat, if you're still on 4.2.2 ; kindly update to 4.3, the heat is noticeably REDUCED.

Like I said, if they know, it may be voided as you installed something not allowed in the T&C.

Rooting on Sony devices I'm not sure... but unlocking bootloader is entirely not allowed.
Besides, only if they are able to access your phone entirely, maybe with terminal (command prompt for Linux) they'll know you have root or not. Without access, how are they gonna find out by just looking at the phone?
I know this cos I just bricked my Z1, sent to them but I hope they don't find out

Chances are slim cos the phone cannot power on.

you're getting me wrong

if they really want to find out in detailed, see under applications drawer also will know ur phone rooted or not cos got the SuperSU or Superuser app. besides, if u have busybox is also another app requires root.

mine is some kind of brick whereby the phone can be seen charging or the PC detects connection on and off, but it cannot be switched on. i've tried charging for 2 days straight, still cannot. so chances are slim unless they have their own tricks up their sleeve to have it switched on again

but usually when you send your phone for repairs (basically warranty), you'll lose your data, which means they'll be flashing new firmware or something over your current to prevent any data clashes.

not entirely a gamble, just that if you know what you're doing, you won't send for repair. i accidentally bricked it cos with my Xperia J, i've never had any issues with flashing/custom ROMs.

brick dy cannot do anything. I can only look at my Z1 physically and drool at its beauty... haha and yeah, just root if you want, but you need to revert back to old firmware for that...

hope the battery life won't be affected with all the sudden increase of features