Flash is working without issues on the Galaxy Note 10.1 2014 edition with it's current firmware based on android 4.3. Well it's not as smooth as on desktop PCs but that was always the case with android flash, although the S-Pen makes things alot more accurate...
I hear with Android 4.4. Kitkat there is only a handful of browsers left that support flash. Which is a shame. But as of now with 4.3 it's as good as ever. You just have to download and install the .apk yourself, because it doesnt come preinstalled.
Phone: Samsung Galaxy S III | Stock rooted
Tablet: Galaxy Note 10.1 2014 Edition Wifi | Stock rooted