Thanks for all your replies, fells good I'm not alone out here with the same issue.
I went to a shop 40 km from here and bought a new Note 3.
Then I started to do some research.
1. New battery - not charging
2. Tried new charger and cord -nope
3. Change USB connector at the bottom of the phone with the new one - still not charging.
4. Tried a new stock ROM via ODIN since I got a new battery - not charging
5. Mother of all..! Changed motherboard with the new phone. Voilá! Charging even with old battery.
Well, it looks like something really bad happened with the motherboard. And I haven't barely used the phone - but rooted it.
I will definitely root again but perhaps with "universal root de la Vega" instead of CF root. Don't really know what's best.
Once again, thanks for your replies and comments. They are worth gold when in trouble.