What is the correct pronunciation of this Korean animal word?

This word is short for the more common 고양이.

While many Koreans do not distinguish ㅐ and ㅔ anymore, the prescriptive standard does distinguish those vowels.

"쥐" is one of the three animal words before which "숫-" comes when one refers to its male. Therefore, "수쥐" is incorrect.

"뱀" (snake) does have a long vowel, but Korean words generally do not have a long vowel in a syllable that is not the first one in the word.

This is because of the 사이시옷 phenomenon.

The general rule is that when ㄹ comes after ㄴ, the ㄴ is pronounced like ㄹ.

In Korean, the pronunciation of a word cannot end with two consecutive consonants.

This word does not include apes.

In the Standard Korean Dictionary, this word means the same as the word 신세계원숭이, thus making "New World monkey" the meaning of the word. However, in the Korean Wikipedia, this term refers to only the capuchin monkeys.

The vowel ㅖ may be pronounced with or without the /j/ sound unless it is part of 예 or 례 (it makes the same sound as ㅔ if the /j/ sound is not made).