Any ideas on my question 2? When I go to WPML > Translation Management and filter on WordPress Archives, the list is also empty, I would expect to see some items there.
The two Toolset WordPress Archives on your website are set to use the classic/legacy archive editor. Only the Archives using the blocks-based editors are shown in the 'Translation Management' section.
For the classic/legacy archives, the string translation method that you're already using works.