I assume your question is 'has Cardbox Server a RESTful interface?'
The free documentation - if you had read it - has an answer for you.
Although: Cardbox Server is fitted to communicate with Cardbox Client and using all functions of it. You also can try making something that can communicate with it. Complex, I assure you! An example of such a thing is Cardweb. That is a web server that talks with Cardbox Server. However, it has its own type of querylines. Also it has some limitations. And you need to reboot every 24 hour to keep it stable.
A better alternative is using Microsoft Classic ASP + Cardbox Client. Let that talk with Cardbox Server and you can query whatever you want. The combination fits perfect because as well as ASP as Cardbox uses VBScript as script language. It is very stable - I programmed several applications with it. Deep linking no problem. Good alternative for GET, POST etc. And if you want to use that, just let your scripts it understand...
For making such a thing, good knowledge of Cardbox Client is indispensable.
From Cardweb, "Cache ages for the demo were set to 24 hours in "cardweb.ini" (24 hours=86400 seconds) and maybe you prefer smaller values. In that case you can adjust the appropriate lines in the section [cardbox1] where settings for the "cache refreshment age" are fixed. When less urgent, a script rebooting the CardWeb server every 24 hours will do best".
It reads as though rebooting is suggested only an easy workaround for less urgent deployments.
Max records for CardBox is 16,000,000 (is that right?)
I am looking for a *nix-hosted DB and HTTP output (REST is no deal breaker). It is mission critical. Security must be bullet-proof. Caching is important to mitigate spikes in CPU load. There can be no reboot. The DB does not need to support relational queries but it must scale. I have an idea on how to build from scratch but looking at what already exists.
The reboot of Cardweb is not caused by the cache. Using it after one day or so, often Cardweb does not interpreted its queries any more. So to be fresh every day, I shut down en restarted (at night) by Windows scheduler the Windows service in the applications I wrote. That was enough to keep it stable and not be faced with a disturbance.
Max records: I think you're right. This size, I do not have experience with it. My max was 1.2 mio records.
Asp has it's own caching. So, don't worry about this. On my site I boot after 60-70 days (http://stortenbeker.eu). I use now since 9 months a win 2012 R2 virtual server in cloud. And after some testing with settings, it feels good now. Cheap and easy!
Cardbox stays an smart and amazing tool. And because it is a little unknown, you met no enemies on the www.... They don't know it! Ssshht.