About Foreign names in Poland

What happens when a person comes from a country that doesn't use Christian/Western names, such as from Africa or Japan? How do they spell their names? I heard in Poland, they have some laws on names, but perhaps that is only for the native-born. How do they spell foreign names?

There are transcription rules for each language, set down by an official body. These rules apply to languages with non-Latin alphabets only, of course. So if an African language is written in the Latin alphabet, the names should be left as they are. Diacritic marks might be omitted for obvious reasons, but this applies to other European languages as well.
I know there are international transcription rules e.g. for Chinese, so the Polish spelling Peking is being replaced by Beijing, but I'm not sure about other languages in this respect.

1/ (Asian)
Well, the language has it's economy, we never needed trascription rules for Chinese of Japanese, because our contatcs were rather very limited (in the past), so there was no need to bother. Now almost everything comes filtered through English (because any Asian business that operates internationally has to use some form of transcription to the Latin alphabet, so they do it by themselves, also I assume individuals must have some latinized verions of their names in the passports, because I can't immagine an American, or British or Polish customs/immigration officers letting those people in when they couldn't even read the person's personal data.), so I guess we'll use those forms, without having to transcript on our own. Although the Polish pronounciation will be often completely different in such cases from the intensions of the Asian part.
For example the spelling Beijing is completely wrong in Polish, we do not pronounce "j" as "dż", so it would be more confusing, maybe "Bejdżyng", but why? We don't need a "new" name for the Chinese capital, I'm sure Pekin will remain Pekin (that's simply a traditional Polish variant, like Londyn for London, Rzym for Rome etc.).

2/ European (Slavic)
The laws learning is talking about apply probably only to Slavic languages with non Latin alphabetes.

3/ European (other, Latin alphabets)
About diacritic marks, they were indeed generally omitted, except maybe for the German "Umlaut" sounds (ü, ö, ä) spelled traditionally as "ue", "oe", "ae"; also this "scharfes s" spelled as "ss" (now in German as well). Today, thanks to the modern technologies, writing/printing foreign letters is much easier, so they aren't such a big problem like even 30 years ago, hence the original spelling (for example stressed vowels in French, Italian, Spanish, Czech, Slovak) will be more common, I guess.

There are definitely rules in place for the official transcription of place names and probably given names as well. My father, as an Orientalist, does on/off work for the Head Cartographer of Poland (yes, there is such a person), and this work consists precisely of agreeing upon or disputing the transcription of place names in South-East Asia (not the Far East, a Sinologist would do that). The popular method of using English or French transcription is both outdated and frowned upon. People in the street might say Pekin, but maps should not. I think this applies both to Asia and Africa.