Cannot find the category for Backwards Compatible games anymore in the Store on my series S

You cannot download the game, as it is not backward compatible on the Xbox Series S. The only Conan that is, is Conan Exiles for the Xbox One.

All games will still have a listing on the store, but that doesn't make them backwards compatible. The games you listed are not backwards compatible so there is no way to make them show up on the Series S. Backwards compatibility list

All One games are backwards compatible with the Series S/X (except for Kinect games). The store's description for the game also notes that it will work on both.

If the game you purchase on the Xbox 360 is backward compatible, list here https://www.xbox.com/en-gb/games/backward-compatibility?cat=xbox360 then yes, the game will show up in your library on the Xbox Series S, and you will be able to download it at no extra cost to you. If the game is not backward compatible, then no, it won't work on the Xbox Series S.

If you look at the list on the link I gave you, you will find that Eternal Sonata is not backward compatible, so the only way you can play the game is by using an Xbox 360.

you would only be able to buy backwards compatible games on the Xbox one/series store. They do not go backwards to the 360.

You cannot buy games once the store has deactivated purchasing, or add new games to your account for 360 store