Address database of France, geocoding and reverse.
adresse.data.gouv.fr API is a free Geocoding API that requires no API key, is served over HTTPS and has unconfirmed CORS support.

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Yes. The adresse.data.gouv.fr API is listed in the Geocoding category of the public-apis directory, which only includes APIs that are free or have a free tier. Check its documentation for rate limits and paid plans.
No. The adresse.data.gouv.fr API requires no authentication, so you can start making requests without signing up.
CORS support for the adresse.data.gouv.fr API is unknown. Test a request from the browser, and fall back to a server-side call if it is blocked.
Yes. The adresse.data.gouv.fr API is served over HTTPS, so it's safe to use in production.