Well if its doing it across different roms, it may be a hardware problem. Did you say the reboot menu comes on? It sounds like your power button is messed up. Is it in a case that may be pressing the power button?
Is your s3 a triband? You can tell if the model number ends in t.
If not, to rule out software, I would recommend using odin to flash the stock nj2 tar.
Make sure you have all important data backed up.
I would never recommend kingroot, it sends your phone info to Chinese servers. All you need for root is to use odin to flash a tar of twrp. Done.
nj2 stock .tar
twrp recovery
Get the newest twrp .tar file. Should be at the top.
After flashing the stock nj2 tar, you will be completely stock. Then, using odin, flash twrp. pull battery when its done flashing twrp, put it back in and boot recovery (hold power+home+volume up)
Select reboot/system. It should ask to fix root, swipe yes. You will then be stock rooted. If your reboot problem persists, then it is likely a hardware issue. I can't really help you there. Let me know how it goes or if you have any questions/problems.
Also, I would stay away from cm13 right now. It's not stable enough yet. A good ROM is PAC lollipop. It's what I'm using right now. I can get you a link if you want. Make sure to use gapps for the right version of Android; I.e.; 5.1 gapps for 5.1 roms, 6.0.x gapps for 6.0.x roms. I'm sure you know that, but I try to be thorough.
PAC ROM