NASA Astrophysics Data System.
NASA ADS API is a free Science & Math API that requires OAuth authentication, is served over HTTPS and supports CORS, so it can be called from the browser.

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Yes. The NASA ADS API is listed in the Science & Math 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 NASA ADS API uses OAuth authentication, so you need to register for credentials before making requests.
Yes. The NASA ADS API supports CORS, so front-end JavaScript can call it directly.
Yes. The NASA ADS API is served over HTTPS, so it's safe to use in production.