My Note 2 won't go into "deep sleep" while its charging. It'll go into deep sleep if it's off tje charger and I'm using it thru out the day. Yes, I'm using the original cord and wall charger.
Use battery monitor widget to log what charging rate you are getting. The stock charger is 2a,which should be able to charge it in less than 3 hours. Even a 1a charger would take less than 4 hours. The only way it is charging that slow is either you have something else draining power or the charging rate is around 500ma
Based on the image posted, something is definitely waking up the phone on a regular schedule - even when the screen is not coming on. Check your account settings to see if anything is syncing every 15 minutes or so (maybe work email?) or you might have photo sync set to only when charging?
I would back everything up and wipe my phone. First check battery monitor widget to check your charge rate. Also try another charger, ask Verizon to test it for you.
Oh god. Do this daily and in less than a few months you'll be staring at burned out images on your screen :-X I mean it's your phone so if you're cool with that... yeah...