SNCF public API.
Railway Transport for France API is a free Transportation API that requires apiKey authentication, is served over HTTPS and has unconfirmed CORS support.

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Yes. The Railway Transport for France API is listed in the Transportation 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.
Yes. The Railway Transport for France API uses apiKey authentication, so you need to register for credentials before making requests.
CORS support for the Railway Transport for France API is unknown. Test a request from the browser, and fall back to a server-side call if it is blocked.
Yes. The Railway Transport for France API is served over HTTPS, so it's safe to use in production.