The game keys can be used once - then they are attached to that Battlenet that you claimed it on.
To download a game on that Battlenet, first download the Battlenet Desktop Launcher then log in to your Battlenet that has the game. The login is email/password format.
The Install button should be in the games tab menus. There will be only one install option. The game contains everything, and the keys on your account just unlock what you have access to.
Also, the base game is free now, and the only way to get expansions now is the Expansion Collection in the Shop. I don’t know if that impacts anything with your issue or not.