That is almost certainly the first book of Jean Molla's series Les Revenants, namely Le sort d'éternité (2006).
The cover features a cat and an evil-looking floor:
Translated from Babelio above:
Quentin's life changes when he moves into a fifteenth-century home and discovers a strange stone in the cellar floor. This stone exerts a harmful fascination on his older brother Nicolas. Soon their neighbor Violaine, an old woman, warns Quentin against the forces of evil and enjoins him to remember his past. Little by little, he realizes that the house belonged to a master sorcerer, Jacques Guernière, burned alive in 1457, and that he is his reincarnation. Faced with Nicolas won by the powers of evil and preparing the arrival of the Lord of the Flies, Quentin begins his fight against the allies of the Devil and discovers his powers as a sorcerer...
Our hero just moved to a small city/village with his mother, I'm sure, a cat, maybe a sibling?
Main character Quentin moves to Comberoumale (hint: typical French isolated town's name) with his mother and brother Nathan. Not sure if the cat on the cover was theirs, but later on in the series a character can shapeshift into a cat.
The young boy finds magic circles, a summoning demon thing, in the cave of their new house
Technically, it was a paving stone in the basement.
His neighbour is/was a witch and teaches him about magic, with a wand I think
Violaine, an old woman, is the one who introduces him to the (not-so) legend of the evil demon Azael sleeping in their basement. She has the "guide" role, providing info on magic, legends and stuff.
There is no "wand" but a "bâton de sorcier", which roughly translates to "sorcerer's staff", passed down by Violaine to Quentin.
Later, the cat kills a rat in the cave, on the summoning circle and the blood wakes up a demon
Azael, the demon trapped in the stone, emitted some kind of smell which attracted the local cats. The prologue features a cat killing a mouse, and instead of eating it on the spot, taking it to the house's basement and eating it there. The blood is sucked by the stone. Here's the extract, courtesy of Google Books:
L'obscurité est totale, le chat trottine vers le centre de la pièce et dépose sa victime sur la dalle froide, aussi rugueuse qu'une pierre ponce.
Pourquoi vient-il ici chaque nuit dévorer les rongeurs qu'il attrape? Il ne sait pas. Il vient. Comme sont venus d'innombrables chats avant lui, depuis une éternité, sans que quiconque le leur ait appris. Sans que les occupants de la vieille demeure - quand ils étaient de ce monde - l'aient jamais su.
L'odeur épicée, métallique, qui flotte dans la pièce l'enivre et avive son appétit. Ses canines déchiquettent le cadavre. Les os craquent sous ses mâchoires, les sucs giclent. Quelques gouttes de sang tombent sur la dalle qu'il ne prend pas la peine de laper. En aurait-il le temps? A peine ont-elles touché sa surface qu'elles disparaissent, comme bues par la pierre.
My English translation below.
The darkness is total, the cat trots towards the center of the room and drops its victim on the cold slab, which is as rough as a pumice stone.
Why does he come here every night to devour the rodents he catches? He does not know. He comes. As came countless cats before him, for ages, without anyone having taught them to. The occupants of the old house - when they were still alive - never knew.
The spicy, metallic smell that floats in room gets to his head and kindles his appetite. His canines shred the corpse. The bones crack under his jaws, the juices spurt out. A few drops of blood fall on the slab that he does not bother to lap. Would he even have time? Hardly have they touched its surface that they disappear, as if sucked by the stone.
Also, Nathan (the main character's brother) did several sacrifices on the stone, allowing the demon to be awoken/regenerated.
The story took place nowadays, I read it around 2009 but the book could be a little bit older
The context was indeed modern, and I read that circa 2010-2011. The book was published in 2006.