Sorry for your confusion and the underwhelming docs. Let’s try to clear things up.
As far as the firmware goes, Github hosts the most recent “official” release. (The one from January you saw in the forum has no additional features.) So, yes, just grab gameFrameWifi_20161211.bin. Make sure you’re downloading properly from Github though (i.e. click the download button on this page).
You’re using the right page for managing your firmware. As for “tasty_station,” that is probably the temporary name given to your Photon when you registered it with Particle and made that account. You should have been given the opportunity to rename it then. You can change it by logging in here and clicking on the name. You can also access and play with functions that control aspects of Game Frame from there.
System version 0.6.0 is the Particle firmware version, not Game Frame firmware. To see whether the firmware uploads successfully, go to http://www.your-game-frame-ip/help and see it listed there. It should say 20161211. (It’s also very easy to tell simply using the Game Frame remote because a third mode is added (FX) in addition to Gallery and Clock.)
As for the dependence on Particle.io, you’re welcome not to use the exposed functions. You don’t even need to login to their server to update your firmware if you want to use DFU mode. I just find it’s much easier to do so. The built-in web backend (your-game-frame-ip.com) doesn’t rely on their services. That said, I’ll probably end up abandoning the web server (or making it opt-in) because it’s too buggy and causes the Game Frame to hang if it loses connection to the Internet. I run a watchdog that reboots the Game Frame in this case, which may be the root of your original problem where it suddenly turns itself on.
Can you describe what happens when it randomly boots? Does it reboot from an off state (powered down using the IR remote)? I may not have considered that case and can add a check during boot to see if it’s recovering from a hang AND was powered down, in which case it would return to an off state. I know this is a workaround, but the web server hang bug is deep in the Particle.io source and I don’t see them fixing it anytime soon. The best solution is probably to drop the web server and expand the firmware manager to have the same features as the back end, populated by exposed functions and variables. At least these don’t crash.