Is there puma risk?
The chance of attack by puma (similar to the North American cougar) is miniscule. You have far greater worries on the Paine Circuit.
Yet, in March 1998, a fisherman was attacked and killed when he disturbed a puma eating its prey on the shore of Lago Sarmiento.
If you are approached by a puma, be aggressive making noise & waving your arms. If attacked, fight back as if your life depends on it. It does.
Storing food safely
At night you should cache food out of reach of animals, including fox & mice.
Most hikers keep all food inside the tent (not the vestibule) but consider the possibility of a rat chewing through your tent to get at the food.
It does happen during summer high season.
Some campsites have wire lines with suspended soft drink bottles. If you hang your food between the bottles, rodents are unable to get to it.
Wash your dishes before going to bed as dirty pots are certain to attract animals.
Are there any dangerous snakes?
No, none.