English and Ukrainian seem to have borrowed this word from Turkish (divan = sofa), and Turkish borrowed it from Persian.
According to wiktionary:
The Polish language borrowed this word from Arabic word divan meaning court or bureau, but in Arabic this word descends to the same Persian word.
So the words in English, Ukrainian and Polish are etymologically related, but they became false friends, indeed.
From wikipedia: https://en.wikipedia.org/wiki/Divan_(furniture)
"Divans received this name because they were generally found along the walls in Middle Eastern council chambers of a bureau called divan."
In 19th century, Turkish style furnitures were popular in Europe, and "all the boudoirs of that generation were garnished with divans."
So my guess is that in English, "divan" came to refer to the sofa part of the furniture set, while in polish, "dywan" came to refer to the carpet part.
I hear the first word spoken as two syllables. I wrote it as one for obvious reasons, at least two me. Should it be spoken with an (ee) sound at the end? And at the end of the last word I hear an (eh) sound. Perhaps this is me, but it is quite frustrating. Is it me? My hearing specialist says that my ears and hearing are fine.
I think your hearing is ok. When you hear 'mój' somewhat like 'mu-i' than it's correct. Normally there are no vowel-connections in Polish like in German "au", "eu". They would be pronounced as two vowels: "a-u", "e-u".
Regarding the "y" at the end of the last word: I think your expectation in a "y" just differs from what it is actually spoken in Polish. It's not a "y" like in English. So ... i assume you hear a correct polish "y" ... :-)
Quite. Polish "y" is not English "i". You could try going to https://en.wiktionary.org/wiki/my#Polish and playing the sound there.
An English approximation is given at https://en.wiktionary.org/wiki/Appendix:Polish_pronunciation where it is said to be like the "e" in the word "roses". English approximations can be tricky though, as two people from only a hundred miles apart might differ quite a bit in their pronunciation!