Hello,
Each of these coins you mentioned has its own wallet. You can run all the wallets on the same server or on different servers, depending on the resources available on your server(s).
To connect to a Bitcoin core full node, use the built in adapter and give the connection information to connect to your daemon.
To connect to other Bitcoin-like wallets, such as Bitcoin Cash, Litecoin, Dogecoin, you must currently use the Full Node Multi Coin Adapter extension.
On your server, you can have as many wallets running as you like, as long as you have enough CPU and disk space. The nodes all listen to different TCP ports and will have different login credentials.
(In the upcoming wallets6 plugin this changes, as you will be able define an arbitrary number of Bitcoin-like wallet connections in the plugin without installing an external coin adapter. All will be explained at the release notes. For now just keep in mind that you may need to reenter your connection settings to reconnect to your wallets. The process is easy, easier than the current process, and I will be available to help you along the way.)
with regards,
Alex