Plain being rooted should not interfere with OTA updates, Kies might not like your device being rooted, but OTA (Over The Air - using Wifi or network) should work OK, it certainly worked on my SGS4. Updating the firmware by any official method will get rid of root.
It is and it isn't ocnbrze. I believe the Note 3 N900 8w is the same spec but has a different modem to the Note 3 (SM-N9005) which we are strictly speaking in this root forum about. The OP must be careful to flash correct firmware for their device.
Daniel, type *#1234# into the dialler and tell us what the three lines of information show exactly. If your rooted this might invoke the super user application, if it does just tap the back soft key. It does sound like your using rooted stock firmware.