Does "composer remove drupal/module" actually uninstall the module first? Does composer remove drupal/module actually uninstall the module first,  especially in  multi-site environments?

On the other hand, does composer require "enable" the given module for all sites?
No, Composer doesn’t install or uninstall modules in the site itself.

It’s a dependency manager; all it does is compute your requirements, then download them.

You might be able to add a post install/update command to automate it, but trying to make it generic would probably be a pain.