Showing posts in both languages by default ?

It is possible to do that with xili-language. In this plugin, language is a taxonomy and can be used as others taxonomies in (future) queries. (far better than custom fields)
One language part is not necessary the mirror of another language part.

For home page, disable browser language selection and sub-selection..

For categories, if url is without ?lang=xx_yy (fr_fr or en_us) both posts are listed.

The best way to progress : test it.

M.