I like the remote code editing feature. I would love a solution that lets me write code on my iPad, including running the code locally. If I could run a server at home and connect to it it would be great. Is there anything like that?

You can't run your code locally, but VSCode (but NOT VSCodium sadly) has a set of remote development plugins (ssh, containers, etc) that allow you to edit locally, run on a remote server.

Also see code-server (https://github.com/coder/code-server) for another option.