I think we are not hitting the nail on the head. The OP keeps stating that the device runs hot.
As anyone with computer repair experience would know is that heat will kill a device.
Overheating = serious lag.
My guess, as others have said is that some app you may have on the device is using the cpu a lot. Hence the heat build up. The heat will also cause the keyboard lag you see, as the device is trying to catch up while the cpu is being pounded by multiple threads.
Think of a desktop computer, in the hardware sense. A desktop can have an 8 core cpu meaning that it can process 8 or 16 instructions per clock cycle. Now, bombard the cpu with 100 processes that tell the computer to divide 22 by 7 (pi for those that know), and carry it to the billionth decimal place. Then when it is done do it again.
The cpu would then put all process in a queue behind the arithmatic in order of query. This means that while the cpu is processing to the nth degree, your keyboard press is on hold for the second or so while the cpu finishes the job at hand.
If there was a light on the front of the phone showing you how your cpu is being used, I would venture a guess that it would always be lit. Could be as simple as a app that doesnt play well with 4.2.2.
Sorry so long. As for my S4, after the update pushed by at&t, phone runs as smooth as my note 2 and my GS3 before that. Camera startup issues (lag) could be attributed to the fact that the camera starts recording as soon as the app loads for the best shot functions as well as some video features. Supposedly, it caches like 3 seconds of video or something for the editing tools to function properly.
It's not wrong to fondle the s pen.