Well, the labels of input fields are written in English yet user enters his name in native language.

What's the reason of having name at all? You can call the person by this name. But if I write you my name in my language, what you (not knowing how to read it) can do? Only "hey, still-don't-know-you, here is your info".

In my foreign passport I have name __transliterated__ to Latin alphabet. Shouldn't this be the case for other places?

w3c recommends adding a separate field for pronounciation - see e.g. paragraph after second image in https://www.w3.org/International/questions/qa-personal-names...