Field collection translation
For the record, if other people have a hard time figuring how the translation in Field Collection works, here's what I found out:

It does not work.

There is an issue here: https://www.drupal.org/node/2618016 that provides a patch to add support, but does not retroactively add translation support to existing field collections. If your site already has field collections you are SOL - and in my case the whole site becomes non-functional.

I'm now looking into migration towards paragraphs, but not sure how to migrate my data. Will update when I have a good solution.