Jerusalem in Hebrew is usually spelled Yerushalayim. Is it Yemenite Hebrew when it's spelled Yerushalaym and Yerushalem?

No

Just bad English spelling.

The phonemes in Yerushalayim don't really change between the main pronunciations of Hebrew.

As opposed to lets say

Shabbat / Shabbis / Shabbath

Hebrew is, of course, written with the Hebrew alphabet (as יְרוּשָׁלַיִם), not the Roman alphabet. (In Arabic letters the same name is written أُورُشَلِيمَ).

All the different transcriptions used when one is writing in English or another non-Hebrew language are just different people's ideas of how the word sounds.

Its spelled the same in Hebrew regardless of which pronunciation is used. All that change in temeni vs Asheknazi vs Sefardi etc Herew is the accent- in written form they are all the same