Sounds to me like a bad battery. The good news is that a good replacement is $16, or you can get 2 and a charger (that can charge the one in the phone and the one not in the phone at the same time) for $25. Look at
Amazon.com: note 2 battery for the Anker (3rd and 5th) items.
Keep the extra battery with you (in a plastic bag, so you don't weld your keys together) and when the one in the phone gets down to about 40%, swap them. You'll be surprised at how long they'll last if you don't discharge them past 40%. (I still use the original batteries in my spare phone - from 2004.)
As far as calibration, that's one of the big misunderstandings about Android. The file being changed when you "calibrate" the phone just keeps track of what percentage of battery use is being used by which app. It has nothing to do with how fast the battery is used or when the phone shuts off.
However ... with any new battery, fully charge it before even turning the phone on. then use the phone until the phone tells you to recharge it (which may be with 5% charge left - this is the ONLY time you discharge past 40%). Fully charge the phone (preferably turned off), then use it again until the phone tells you to charge it. Go through 3 cycles of full charge, full discharge. It won't hurt the battery (it'll shorten the life by a few days at most - out of a few years), but if the battery has been sitting on a shelf for a few months, it'll condition the battery so it'll last years longer.
BTW, rooting the phone has about as much to do with the battery as it has to do with how fast weeds grow in your garden. All rooting does is install a file (or two, depending on what you use to root it) and one app. If you don't use them, they don't have any effect on the battery. If you do, they each run for a fraction of a second and use less battery than having the screen flash on just long enough to see it. Well, I guess rooting
does have more to do with the battery than with weeds, but not very much more.