1. Your Gamertag is connected to Xbox LIVE not the Xbox 360 or Xbox One, I have my Xbox 360 Hooked up to my TV through HD cables and my Xbox One connected though HDMI and I am able to alternate between the 2 logged in to Xbox LIVE the entire time. I don't know if you could play both consoles on an online game at once though I've never tried that.

  2. Yes. Xbox Live is tied to your Gamer Tag.

    Keep in mind, you can only have one account logged in at a time. So, if you are playing your Xbox One, someone else cannot use your login to play on your Xbox 360.

    Hope that helps

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