Technology would seem to be the perfect place to post!
It's getting harder and harder to get hold of multi-region DVD players (with or without all the other features you're seeking) these days.
The licensing arrangements that manufacturers and distributors have to enter into have always prohibited the sale of multi-region players in order to ensure that film companies can maximise their profits by selling the distribution rights separately in different parts of the world.
In the early days of DVD players though, manufacturers made all of their machines as 'multi-region' and then just applied a bit of coding to them at the end of the process to restrict them to use in a particular region. That coding could usually be easily removed by following widely-available instructions on the internet. (Companies such as Richer Sounds would advertise multi-region players but then supply them as Region 2 only, in order to comply with the rules, but include the instructions for removing the region-coding inside the box).
However the movie companies got wise to that and have forced manufacturers to 'hard code' their players, so that the region-coding can no longer be removed by applying a simple hack. So it's becoming almost impossible to get hold of multi-region machines.
DVD recorders are also getting harder to come by these days, as recording DVDs to time-shift TV programmes has become 'old technology' since PVRs (with hard drives) have become available.
So you'll struggle to find something to match your exact requirements. The closest match might be the Panasonic DMR-BWT850EB though. It's not multi-region (which I think would be an 'impossible ask for such a machine anyway) but it can record to a massive 1TB hard drive, as well as to DVDs and to Blu-ray discs:
Amazon.com User Recommendation