I know next to nothing about stock Roms, because I had rooted my Note the second day after the purchase and never looked back.
Still, I don't think that your heating issue could be attributed to running stock kernel alone; it's more likely the kernel/app combo that's bugging you.
Stock Roms are notorious for being lumbered with tons of crapware that behaves the way it wants, not the way you'd like your system to behave.
Rooting your stock Rom properly (not with CWM.zip) with a decent TW kernel could be the answer to your problem: you'd get a well-tested kernel and you could remove various Samsung faeces like music and social hubs etc.
I'd recommend using Notecore kernel as a first step towards resolving your problems. Go to the OP of this thread and download Standard NC version for PC Odin - it's a tar file.
1. download PC Odin if you don't have it (Google PC Odin 1.87) and run it from your PC
2. click on 'PDA' and browse to the Notecore tar
3. put your Note into download mode (volume down + home + power)
4. connect the phone to the PC
5. wait for Windows to install the necessary drivers
6. wait for PC Odin to display a yellow highlight with a COM number
7. hit flash
8. you are properly rooted and running the Notecore kernel
It shouldn't take more than 4-5 minutes. If all goes well download
Triangle Away, install it and reset your binary counter.
Then you can proceed with removing Sammy crap and tailoring your Note according to your needs.
Important: Do NOT use your existing CWM to flash anything - it is dangerous. After flashing Notecore you can use its CWM to do whatever you like.