How do i actually sign up for xbox live and stay signed in?

Hi John-Logan, my name is Rodrigo. I'm an Independent Advisor and a member of the Microsoft community and I'd love to assist you today.

I am sorry to hear that.

Please, follow these steps:

Press the Guide button on your controller, go to Settings, and select System Settings.

Select Storage.

Highlight but don’t select a storage device, and then press the Y button on your controller for Device Options.

Note It doesn't matter which storage device you choose. This clears the cache for all storage devices.

On the Device Options screen, select Clear System Cache.

When prompted to confirm storage device maintenance, select Yes.

Source: https://support.xbox.com/help/xbox-360/console/clear-system-cache

how do i sign out of xbox live while staying in my gamertag?

Hey mate, you will need to use the 'Recover Gamertag' option to use that account.

HELLO I JUST SIGNED UP FOR THE FREE MON. OF GOLD... HOW DO I LOG IN?

(use your hotmail or live account that you sign up for and sign it up on your xbox

I've had multiple codes:

800704CD

80072746

D0000034

8007065B

80151000

They pop up when I try to connect or when I get disconnected. I can actually sign in xbox live with my profile, on my cousin's xbox, at my place. I can actually sign in with my xbox, using his hard drive. I can stay connected with my HD and my xbox as long
as I don't sign in my profile. My cousin can stay signed in his profil on my xbox and my HD and it works fine.

The problem seems to be MY profile, only on MY console, only with MY hard drive. I will probably try to buy a new hard drive when I have some spare time but it still stucks.