I thought I'd add my experience. My wife and I both have note 3's, purchased at the same time. I've had zero issues with mine, but my wife had been complaining about screen locking up recently and not being able to swipe, but using the pen works. So I took a look at her phone and played around with it a bit, and for the first time I saw what she was complaining about. Swiping through the home screens the screen locked up half way, and literally stayed like that until a good few 3-4 seconds. I brought up running apps, but the odd thing was that screen was stuck, as I couldn't scroll thru the entire list or close anything down. Camera app, fb, svoice,and some other apps were on there.
Configuration wise we're fairly identical, although I noticed she had Smart Stay turned on... I've turned it off since.. but still not sure what the actual culprit was. Still a bit concerning.. Warm battery? she tends to leave it charging quite a bit. i'll try to monitor it a little more.
As far as general lag is concerned, I haven't really seen any. only time was when on my phone I somehow had screen mirroring enabled if I recall correctly. A lot of screen updates were a noticeably laggy.. but turned off its smooth as butter. Odd thing is, now when I try to replicate it I cant seem to enabled it without syncing against a device -- so im not sure how I had it enabled previously...