do you have any intention of adding language support to the API methods for people (for biography, etc.)

Yes, this is planned although not on my immediate list of things to do.

movie casts (for character name)?

Haven't heard anyone request this yet. Are there really many movies that would have different character names in different languages?

It does happen, though I'm not sure how often. I suspect it's probably more common in films based on books, where the character names were changed in the translation of the book for whatever reason, and the film translation follows the book. You also get local versions of common names (e.g. "George" to "Jorge") or titles. Unnamed characters sometimes have descriptions (e.g. "woman in elevator"). Languages that don't use Latin characters will have to use some kind of transliteration of names. There are probably other reasons.

I'm not particularly interested in having casts translations. I was just wondering if you were considering it.

I'm encountering a problem with mixed up languages, when naming characters. They are sometimes in French, German or other languages. (e.g. His Wife, Seine Frau, Sa femme) - Is there a guideline, which language to choose? Should it be in english or in the native language of the movie?

Well, its not that bad, but i really don't like inconsistencies…

One of the ugliest examples i found: https://www.themoviedb.org/movie/29267-der-m-de-tod/edit?active_nav_item=cast

Many different characters played by each actor and some translation additionally added.

How do I at least get the existing person biography in a specific language? For instance when I query Sophia Marceau (person/1957) her biography is returned in French. Adding the "language=en" parameter doesn't help. Thanks.

Person biographies are not translatable at the moment. It is assumed that every biography is in english. if you find one that isn't, you should change it.

Could you still implement language support for person names (crew members and casts) and also for 'Production Companies', please? Because I would like to see person names in my native language by my native letters. Thanks.

Translated people (names/biographies) are coming in the next few months but I haven't committed to when I'll work on translated characters. I have an open ticket for this here.

Cheers.