I don't think VS/GH Codespaces are open-source, and they don't seem to support GitLab.

However, if you do need something based on open-source that works with GitLab, take a look at https://www.gitpod.io

You can even self-host it: https://www.gitpod.io/blog/gitpod-self-hosted-0.4.0/

It used to be called Visual Studio Online (it was renamed to this) and you could indeed run it on your server, so I'm assuming that's not going to change.

No clue about GitLab integration. I don't see myself using a browser tab instead of an app to write code, and the app does have GitLab plugins, so I'm gonna go with probably.

The Visual Studio Codespaces have coverage of using your own server at https://docs.microsoft.com/en-us/visualstudio/online/how-to/....