Monkeys of course did not inhabit the Italian peninsula 2000 years ago, or anywhere else on the north end of the Mediterranean. But they did exist in Africa, and the Romans (and the Hellenistic dynasts before them) were keen to import exotic animals. The Romans, in particular, had a voracious appetite for exotic beasts to kill in the games (the animal hunts, venationes). When Cicero was serving as governor in modern-day Turkey, his young protege Caelius Rufus frequently asked him to send leopards back to Rome for his ambitious programme of games (Cicero never seems to reply or comply to this). Small animals were also common as gifts between young lovers, as a subtype of exotica along with perfume, precious stones, figurines, et cetera. We even find some burials of pet monkeys with grave goods, like the Macaque found at Iulia Libica in the Pyranees!

Venomous snakes were everywhere and still are. Pit Vipers still lurk in the closed-up tombs of the Mediterranean countryside today, though I've heard they are less common in Italy now compared to Greece. I don't have any specific knowledge on the commonality of specific species.