Hi, I'm Carl and welcome to Just Answer. The first thing I read was that you have a 1st Gen Apple TV that won't accept your Apple ID. If it's connected online then the login is most likely incorrect. Goto icloud.com and try signing into the same account using any capitalized letters that the password ***** contain. What works there should also work on the Apple TV. Be sure if any letters are caps that you input them as caps when you are signing in on the Apple TV.
If you need any clarification or additional help, please feel free to ask me.
Are you able to signin to ATV using your Apple Account? It shouldn't give you a code unless you are signed in.
It won't show it as a device until it successfully signs in on the ATV. Are there any capital letters or special charters in the password? If so, change the password ***** just lowercase letters and numbers and try it again. If it's connected online it should work.
Ok, I know in the past that the 1st & 2nd Gens would have issues not transmitting the characters correctly using the ATV Remote. Do you have another iDevice like an iPhone, iPod Touch, iPad, etc? If so, download the Remote App and use it as your remote to enter in the Apple ID and Password.
Also, make sure 2 step verification is turned off on your Account as this can also cause a problem.
Let me know how it goes. Fingers crossed! :)