Finally got data working on Galaxy Note 2, like-new refurbished, genuine, rooted, by unchecking and then re-checking "Can access the Internet".
I had to perform a "hardware factory reset", a.k.a. "hard reset", in order for "System Update" to work without crashing Settings, possibly because of rooting though I didn't test it before rooting. Rooting survived hard reset.
After hard reset I did Update PRL and Update Profile as suggested by support, but still no data. :( With no other ideas I randomly tried unchecking and then re-checking "Can access the Internet", which triggered a second PRL update notification. After this PRL update data finally worked.