There's also some fairly comprehensive documentation on the github wiki, including an architectural overview, and building your own printer: http://github.com/freerange/printer/wiki
I suspect that BERG have much grander plans for BERG Cloud than this, and of course BERG Cloud will be much more polished and suitable for normal people. But that said, it's a tool that might help explore a similar 'space' of possibilities, yeah.
You definitely could, and it would be much easier to write the HTTP client code on a Raspberry Pi than it is on the Arduino!
If you look at the Arduino code (https://github.com/freerange/printer/blob/master/printer.ino), all it does is download a stream of bytes, doing some very rudimentary validation (Content-length), and then sending the bytes out on a serial port.
Send a message or a picture using the demo apps
http://printer-mail.herokuapp.com/send/freerange
http://printer-paint.herokuapp.com/send/freerange