My guess is that there would be at least one passenger or crew with a mobile device not in flight mode. If this were the case, then could it's signal be tracked as the aircraft flew along and would the data be kept on file with telecommunication providers?
In theory, yes. The phone may try to perform a location update, especially if it senses a new network such as at a border crossing. But (and referring to e g China Mobile statements) if the location update request fails the home country network will probably not be notified. Most such requests probably fail due to weak radio connection.
In this case there may be a trace recorded in the visited network's database. But since such unsuccessful requests are from the visited network's viewpoint just nuisance data taking up storage space, it is doubtful if it's stored for any length of time. Network operators probably will not make public exactly how this works in each network, but it is conceivable that the accident investigators can be allowed to sift through this data if it exists.