Yep, you can do it...
I do the same thing for my heating system. It tells me what zone is calling, what the temps in each room are, outside temp, DHW temp and re-circ control and sends a txt if there is a low temp alarm. All is displayed via the Ethernet shield as a web server.
As mentioned the limitations fall on the display end and the size restrictions. I have not managed to kill mine yet but I have also no implemented every feature I would like. Having the Arduino do everything is nice since it becomes all you need. But if you can add an external web server with PHP and MySQL, you can do what ever you want on the webpage. Another advantage to having the Arduino talk to an external DB is you can have a script on the server check to make sure the Arduino IS checking in and that is has not lost communications. Last thing I want is a problem with a lift pump! 