Xbox Live not working in a Backward Compatible game (XBOX 360 game) on XBOX One

Any Xbox One games will work and any original Xbox/360 games that are backwards compatible will work. You can find a list of backwards compatible games here - https://majornelson.com/blog/xbox-one-backward-compatibility/

By classic, I'm assuming you mean Xbox original, if so, then if you are using a Xbox 360, then make sure the games are backward compatible, by checking this list on here

https://support.xbox.com/en-GB/games/backward-compatibility/play-original-xbox-games-on-xbox-360 and make sure your Xbox 360 console has an official Xbox 360 hard drive installed as well, otherwise they will not work.

If you are trying to play the games on a Xbox One, then make sure the games are backward compatible, by checking this list

https://www.xbox.com/en-gb/xbox-one/backward-compatibility If they are not on the list, then the games will not work.

That game in particular is not backwards compatible, so it wouldn't work. You can view what games are backwards compatible on the 360 here and compatible games with the One here

IF Skyrim is the Xbox 360 version of the game, then that will not work on the Xbox One as the game is not backward compatible. If you want to play it, then you will either have to use an Xbox 360 or purchase the Xbox One remastered version of the game.
The full list of backward compatible games can be found here
https://www.xbox.com/en-US/xbox-one/backward-compatibility

First off, there is no Xbox 2. there is Xbox, Xbox 360 & Xbox One. Xbox One games will not work on the Xbox 360. Games marked Xbox will work on the Xbox 360 only if they are backward compatible, and your Xbox 360 has an official Xbox 360 hard drive installed
in the console. Full list of compatible games here https://support.xbox.com/en-GB/games/backward-compatibility/play-original-xbox-games-on-xbox-360