After looking at the CSS and template code, and searching more on the forums, I found mention of a plugin that puts languages and currency in the header:
https://www.zen-cart.com/downloads.php?do=file&id=30
Support thread:
https://www.zen-cart.com/showthread....inks-in-Header

Based on that, and my own suspicions, I guess the simplest way forward is to create new divs with id's, write these into the templates/<CUSTOM>/common/tpl_{header,footer}.php as appropriate using logic to only write them when mobile and/or tablet mode is detected, and then control their display with responsive_default.css (make hidden by default), and responsive_{mobile,table}.css (make visible here).

Whether I can merely include the language sidebox there remains to be seen, but I guess not if it contains mention of divs which will be hidden in mobile and/or tablet mode. In that case, I can either change the sidebox code to be more flexible, or simply write the relevant language code into the above header/footer templates.

I could also of course install the plugin mentioned, and use that, if it proves to be a better overall solution.