I can connect them directly to the console?

There is only one completely wireless headset for the 360, the
Tritton Warhead. The Warhead includes wireless sync ability with a 360 console (same as a 360 wireless mic), allowing it to be completely wireless. All other headsets use a cable from the controller to the headset to provide mic support.

If it's a game, it can be played on the same console. Avatar items are the only things directly connected for use only with the gamertag that downloaded them.

Unfortunately no. Games can only be downloaded directly on the console itself. You would have to use an internet connection to get them digitally.

You can make these changes and save them directly from the xbox site, this would have to update the console data when connecting to the internet.

Yes it's the house symbol, I've tried a different Ethernet cable still nothing and both adaptors are plugged into the mains socket on the wall, I've synced them and re-set them, still nothing. "If this light is off when you connect your
console to this adapter, what happens if you switch cable and connect your computer instead?"

My PC is connected directly to the router, I can't connect the X Box directly to the router as it's 12ft away from the router if this is what you mean? Is there's anyway I can connect directly to my PC or a way to bypass the adaptors?