Then for the slug translation, I need to keep the original/default language slug for every language, so I guess I just don't have to do anything there.
Welcome to the WPML support forum. I will do my best to help you to resolve the issue.
The default language Post Types and Taxonomies slug will be adjusted to the secondary language automatically. If you wish to translate it in the future, you can do it from WPML>>Settings. Please refer to this article for more details and let us know your feedback - https://wpml.org/documentation/getting-started-guide/translating-page-slugs/
I have decided to use default WPML languages instead of creating my own, and use redirects, to save the old links. I will redirect /en_US/ pages to /es/ with a re-direct rule. This seems to be a more elegant and easy solution.