Here's the thing that may have confused you. The diphthong ㅢ is pronounced "ui" like saying ㅡ "eu" and ㅣ"ee" in quick succession, but is sometimes pronounced "wae" like in this sentence. Everything else sounds the way it should, double-check with another source if you want.
Shouldn't it say 제 because 저의 changes to 제? It's not part of the rule of pronoun + 의
I don't think so, no. The particle ~의 is to show possession. If you just said 제, it would translate to " I island "
Same difference as 저는/제가 and 나는/내가. 저의 is more formal while 나의 is more casual.
섬 or 도? Because, when you talk about islands, you would say 제주도, right? Not 제주섬. Or what?
While the 도 in 제주도 can mean island, 도 in that sense is more like island in an administrative division, like an island state. 도 could also be referring to province. The Jeju Province, like 경기도, 강원도, etc. Jeju specifically you'll hear more oftenly called 제주도 instead of 제주섬. But for the general term for island, 섬 is preferred.