It has everything to do with index services & sd card. I recently upgraded to Note 3 - and it froze up within minutes of leaving the store. Thinking, perhaps, that a firmware update was imminent, I worked with it but the freeze and lag times increased in both frequency and duration. I returned to the verizon store, and thankfully it acted up while there so no question about the exchange. The new in box replacement worked flawlessly at first, but within a couple of hours the freeze up and lag commenced plus a lot of heat near battery. Not knowing cause, I googled problem and found this thread - so thank you for reporting your issues. And somewhere in this thread - someone mentioned "index service", and how samsung treats that operation. Well, samsung index service is much like Windows index service, it scans all drives to optimize placement of material for ease of access - probably a more technical description exists, but this works for me. Another mention concerned SD card. I pulled my 16gb sd card, and - viola - go figure, the Note 3 came alive and worked FLAWLESSLY, continuing to work as expected. This morning I put in an empty 4gb sd card, no problems. No lag, no freeze, Note 3 is a beautiful and fully capable work of art. As for apps, in addition to factory settings, I have added Fitness Pal, Extensions, Calculator +, Android Sensor Box - but temperature feature doesn't work on Samsung - so this app going byebye. And - no more heat - back of phone is cool to the touch. Conclusion - first unit out the door was working quad core overtime through index service as my 16 gb sd card was 3/4 full of pictures and pdf docs - and likely there is a corrupt pdf on the card, which Note 3 could not overcome - and likely put index service into a continuous loop accounting for lag, heat, and freeze.
My suggestion is to remove sd card, first. Reboot, and run the phone without the sd card in place - to start. From my experience, changing the sd card is answer. Further, you can transfer old sd card information to computer, then transfer to phone if those files are needed. I am doing that now in small increments to ensure hassle-free operation. I hope my information is helpful, and look forward to reading your results.
Cottco