For (1), the logs are supposed to be updating live, unless the scripts are finished. In the case where they are finished, it just shows the output of the log files from when the commands ran. Are you seeing a delay or just not updating at all? Is it possible your script finished or is somehow hanging indefinitely with no output?
For (2), I don’t think pinning the version is supported in devcontainer.json. It always downloads the latest version from the marketplace based on the extension id.
For (3), VS Code is working on getting hot-reload of extensions to work in the web like is already does on the desktop. So in the next month or so that will hopefully work automatically without needing to reload. You can always reload the browser manually today.
It might be easier to start new posts if you run into additional issues so that each post is scoped to a specific problem when other people search for questions/answers