Battery is as usual smartphone - one day or even less if phone is intensively used.
Lag is noticable on lolipop. I dont have experiance on Kikkat, as My D855 had lolipop preinstalled. I use intensive background tasks, so that is, most likely, the cause of it.
Device heating up...? Yes.... More than flagman phone should. Not even on that intensive tasks, but as I sad it could be because of My background tasks.