They do if they fly to Europe, yes. Do they actually use it? Of course not They fly waypoints in the FMS and
AFAIK the FMS doesn't use NDBs for corecting the gyros. But an ADF has to be carried.
True as a generic statement.
However, exemptions do exist, for example US Airways A330s which don't carry ADF but have permission to make non precision RNAV approaches in to Gatwick based on the NDB procedure. Presumably there are others too.