Do lions exists in Westeros or in the ASOIAF's world?
Yes, we hear of a lion in Westeros is Sandor Clegane telling her about the time his grandfather saved Lord Tytos Lannister from a lioness. Note that this lioness is also living in the Westerlands.
"I like dogs better than knights. My father's father was kennelmaster at the Rock. One autumn year, Lord Tytos came between a lioness and her prey. The lioness didn't give a shit that she was Lannister's own sigil. Bitch tore into my lord's horse and would have done for my lord too, but my grandfather came up with the hounds. Three of his dogs died running her off. My grandfather lost a leg, so Lannister paid him for it with lands and a towerhouse, and took his son to squire. The three dogs on our banner are the three that died, in the yellow of autumn grass. A hound will die for you, but never lie to you. And he'll look you straight in the face." He cupped her under the jaw, raising her chin, his fingers pinching her painfully. "And that's more than little birds can do, isn't it? I never got my song."
A Clash of Kings, Sansa II
There are various other references to lions in The World of Ice and Fire but to name one is of the Children of The Forest using cave lions against the First Men.
The hunters among the children—their wood dancers—became their warriors as well, but for all their secret arts of tree and leaf, they could only slow the First Men in their advance. The greenseers employed their arts, and tales say that they could call the beasts of marsh, forest, and air to fight on their behalf: direwolves and monstrous snowbears, cave lions and eagles, mammoths and serpents, and more. But the First Men proved too powerful, and the children are said to have been driven to a desperate act.
The World of Ice and Fire, Ancient History: The Coming of First Men
There is also mention of "Hrakkar" a breed of white lion that lives in the plains along the Dothraki sea.
"Yes, my sun-and-stars," Dany said. Drogo would take his bloodriders and ride in search of hrakkar, the great white lion of the plains. If they returned triumphant, her lord husband's joy would be fierce, and he might be willing to hear her out.
...
The brazier was cold again by the time Khal Drogo returned. Cohollo was leading a packhorse behind him, with the carcass of a great white lion slung across its back. Above, the stars were coming out. The khal laughed as he swung down off his stallion and showed her the scars on his leg where the hrakkar had raked him through his leggings. "I shall make you a cloak of its skin, moon of my life," he swore.
A Game of Thrones, Daenerys VI
If so, WHY is it the House Lannister's sigil?
There's no explicit mention as to why but a Lion is a symbol of great power and strength, something which House Lannister clearly tries to replicate. Also consider one of the possible stories for how the Lannisters came to be.
Other tellers prefer other versions of the tale. In one, Lann uses the cleft to fill the Rock with mice, rats, and other vermin, thereby driving out the Casterlys. In another, he smuggles a pride of lions inside, and Lord Casterly and his sons are all devoured, after which Lann claims his lordship's wife and daughters for himself. The bawdiest of the stories has Lann stealing in night after night to have his way with the Casterly maidens whilst they sleep. In nine months time, these maids all give birth to golden-haired children whilst still insisting they had never had carnal knowledge of a man.
The World of Ice and Fire, The Westerlands
It's also said that Casterly Rock looks like a Lion as darkness starts to creep in.
Casterly Rock, the ancient seat of House Lannister, is no ordinary castle. Although crowned with towers and turrets and watchtowers, with stone walls and oaken gates and iron portcullises guarding its every means of egress, this ancient fortress is in truth a colossal rock beside the Sunset Sea, a rock that some say looks like a lion in repose when the sun sets and the shadows fall.
The World of Ice and Fire, The Westerlands: Casterly Rock
Shouldn't they have picked some animal natural to their geographical region or something like that?
The lion is natural to the Westerlands and is even known to have lived in the Rock itself.
The Rock has been a habitation for men for thousands of years. Before the coming of the First Men it seems likely that the children of the forest and giants made their homes in the great sea-carved caverns at its base. Bears, lions, wolves, and bats have also been known to make their lairs within, along with countless lesser creatures.
The World of Ice and Fire, The Westerlands: Casterly Rock