I have no idea what UK law is. In the US the data itself is public domain, but the compilation is of data is copyright. Maps commonly would intentionally have errors in to detect copying - the error is creative work and so a copyright violation to copy so if someone copies your map you can sue them for copyright violation for not just the errors but also that compilation. If you take someone else's map and then use that to create own map off of (thus finding and fixing the errors) it is legal, but that is as much work as just creating a map from scratch.
Not by default, at least in the US. The database has to actually be more than just a compilation. It's not a high bar to clear, but it's there. Europe and the UK have the "sweat of the brow" doctrine however.
We have a database right in the UK, originally derived from EU legislation. That would apply to postcodes AFAICT, but government can legislate in favour of the demos and against a private corporation.
