Thanks for the reply! I had already given up hope and assumed it was bugged or something, but after I read your reply I logged in and went back to the island, but it still wasn't raining, so I logged out there and went to sleep. But when I logged in today it was actually raining already! I had no idea how long it was going and how long it was gonna last, so I started looking for baby monkeys as fast as I could, and the first one I found was a rare one! However I wasn't too happy about its stats so decided to look abit more, and to my surprise about 80% of them were rare...I think blizzard made the chance for them to be rare much bigger then other pets, because otherwise getting a rare one would be a real pain. I eventually did get one with better stats, so pretty happy right now
And yeah, they do despawn the moment it stops raining, I was about to battle the last one and it despawned a few nanoseconds before I could click on it, so you have to be superfast.