It is quite possible that even a Pi could handle this. So maybe a nano-PC of some kind if you want something more powerful than a Pi3 or equivalent.
Well that is very subjective. The most experience from this community is from Rasbian which is derived from Debian. Dave's Pi installation script is likely to work on a Debian OS. Ubuntu is also derived from Debian so that should also work well. Debian is stable, well supported and relatively light weight; however, it doesn't update very often - of course that is likely to be of benefit to you giving you a stable platform but it is likely that it would mean you would need to take more active charge of some key services such as installing and updating newer versions of Node.js. You wouldn't have those issues as much with Ubuntu.
If you don't need the desktop, either don't start it or chose a variant that doesn't include it.
If you end up with a platform with sufficient spare capacity, I would personally also recommend installing
webmin. This gives you a web interface for managing Linux. In particular though, it gives you the ability to configure it such that it will auto-update Linux software packages for you which can be a reasonably large overhead for running Linux servers.