If you want to track whether the Game Frame is turning on or completely rebooting, you can visit this page and login with your Particle credentials.
Then reboot the Game Frame. You should see the log start to tick by. You can leave that up for a couple days and see what you see. If it actually crashes/reboots you’ll see it in the logs with “recover” in the data column. It says “boot” if it boots normally. That’s a little event that’s published on startup to try to troubleshoot this very issue.
I’m glad you found the HTTP commands, but as I said (and you’ve discovered) the web server is unstable and poorly supported by Particle. You’re much better off using the exposed functions to achieve the same things. They’re rock solid (independent of the web server), and just as fast, but you are using the Particle infrastructure.