I'm finishing a master's degree on paleontology with a paper on Quaternary mammals and while it is centered on Europe I've read also on the Middle East in the past, and you'd be surprised at the kind of animals that turn up in Holocene sites. It's the "modern", post-glacial world, but at the same time there are many contrasts with the range of modern animals. Especially many large mammals that have been driven to extinction or near extinction by human action. Nevertheless, I find Zooarchaeology, the study of animal remains in human archaeological sites, a subject as fascinating as neglected.

AFAIK monkeys are only mentioned in the Epic of Gilgamesh in those new lines of Tablet V deciphered a few days ago. They are an indicative of how exuberant and exotic is this land far away that Gilgamesh and Enkidu travel to, which is not meant to be in Mesopotamia. It is not surprising that ancient Mesopotamians knew about monkeys despite Mesopotamia being devoid of them, because there was trade between Mesopotamia and India from a very early date, and several species of monkeys exist in India.

Regarding lions, they are a common motiff in Mesopotamian art and they are mentioned several times in Mesopotamian texts, not just the Epic. This is not surprising because lions are actually indifferent to climate, and their two main limitations as superpredators of open spaces are the presence of thick forests and a lack of medium and large ungulates to prey on. Lions lived all over the Muslim world just a few centuries ago and only began to trully disappear from it after the 1600s with the introduction of modern firearms, that depleted not just lions but more critically the antelopes, deer and wild asses they depended on. The last wild lions in Iraq and Turley were killed in the 19th century and the last lion in Iran was seen in the 1940s. Today, lions still live in the wild outside Africa in the Gir Forest reserve of Western India. These are Asian lions (Panthera leo persica), a slightly smaller, less social subspecies of lion that can also live in more forested areas than African lions. It was this subspecies that was found in Ancient Times also in the Middle East and even in South-eastern Europe.

Finally, the Epic of Gilgamesh was composed at the tail end of the Holocene Climatic Optimum. It was a slightly warmer time than today, but most importantly, it was more humid. Asian moonsoons left rains all over the Middle East and even deep into the Sahara (nowadays, their furthest influence is in Oman and Yemen). While there were deserts, some other areas that are deserts today were prairies and there were more forests along the banks of the Tigris and Euphrates. In those forests (well, the ones that were not cleared for agriculture by humans) one could find forest animals like wildboars, aurochs, Persian fallow deer and even Asian elephants, while more open, dry areas were occupied by Arabian oryx, goitered gazelle and Syrian wild ass. These in turn were prey of large carnivores like wolves, stripped hyenas, leopards, lions and Syrian brown bears.

As you will find if you google any of these animals, you'd find that most of them are extinct in the Middle East or critically endangered because of human persecution and destruction of their habitat. So, could they exist in the Middle East even with the slightly drier climate? My opinion, given that most were still there after the end of the Holocene Climatic Optimum and were exterminated by human action is: yes, if humans let them.