Slovak Documents for Gender X Applicants

Hi all! Writing with a question about Slovak documentation for gender X applicants. Let's say that you have gender X on all IDs (birth certificate, passport, etc.) from your country of origin, and you've been granted Slovak CBD.

How do they decide what gender goes on your new, Slovak documents?

Do you get to pick between M and F?

Does someone else pick it for you based on whether you look more to them like M or F?

Do you have to provide additional documentation showing a legal binary sex (from country of origin) or medical binary sex? What if you are unable to provide either, or different medical tests (e.g., anatomy vs karyotype) show different things?

If you fill out all forms picking one binary gender, and your official documents say gender X (but otherwise match in name/age/etc.), what issues might you face (if any) when they review your forms?

Thanks for any insight you can give! There doesn't seem to be much information about this out there, or at least an easy way to find it.