Note 3 is no problem!
I have a N9002 for about 1 Week now and itīs working good!
It took me only about 3h to root, install Google apps including PlayStore, and the Nexus Keyboard to get more languages.
All you need you can find at XDA!
I took the N9002 because I need dual Sim AND need to run it all around Asia and Europe.
For the dual sim version you will find nearly no custom roms, but thatīs not neccessary for root, Google Apps and Keyboard!
Only the China Unicom Version works in Europe and most Asian countries!!!!!
I donīt know why you had so much problems with the Note 2?
I had the Note 2 (also dual Sim version from China Unicom) for around one year and it worked perfect. Also rooted and injected Google Apps and Nexus keyboard.