"I live in a country which is rather obsessed by paperwork ... There must be multiple insecure copies of almost everyone's identity documents spread around the net, a lot of them stored on e-mail servers in other jurisdictions,,,
Is there any place that gets this right?"
I think there are a number of principles which need to be built into data protection but seem to be absent:
1. Data subjects' rights are treated in the same way as consumer rights: any contract wording which purports to waive them is void. This is not restricted to consumer transactions. It applies in all circumstances, e.g. sharing data with an employer.
2. Drop all the verbiage about data controllers and data processors. There are two roles, data subject and responsible data holder. The responsible data holder is responsible to the data subject for the safeguarding of the subject, for their own conduct and that of any third parties with whom they share data. The responsible data holder is, in the first place, the party with whom the data subject shared their data. If the original data holder ceases to exist any third parties with whom that data holder shared data become responsible data holders.
3. The data holder is responsible to the data subject in a court in the most appropriate jurisdiction. In the first place this would be either the jurisdiction in which the transaction requiring the data sharing occurred (not necessarily that governing the transaction itself) or the jurisdiction of the data subject's normal residence. There would have to be good reason for the court to be in any other jurisdiction. The data subject's preference would be accorded most weight in such a decision.
4. The data protection registrar would be empowered and required to make unannounced data protection audits on any organisation in their jurisdiction, irrespective of any requirement to register.*
5. The data protection registrar would be empowered, if they believed the regulations were being breached, to order a data holder to suspend processing personal data and require any third party to whom they had shared data to suspend processing that data.**
* This, I think, would deal with such casual holding of data the OP mentions. That would provide a disincentive for departments or individuals to hold data contrary to company policy as it would very likely make it a sacking offence within the company.
** A similar provision was in the UK DPA 1.0; it was obviously a powerful sanction but I believe it also intended to act as a trip-wire in that it could be applied without having to go to court. Court proceedings would result in arguments and delay. It would be a last resort but could be applied at once after which continued processing would be an offence in itself resulting in a prosecution which would be very difficult to defend even if the DPR's original view could be successfully argued against. At least this was my reading of it; at the time the DPA was past my job was very much involved with the courts and this aspect of it struck me as likely - in fact obviously - being intended for use in this way should the need arise. I never heard of it being invoked as it was preferable to educate and, if need be, chivvy organisations into complying.