Suddenly, you notice an unusual movement in a birch standing by the road.

"Standing" is technically correct, but in terms of trees, is typically used in a more poetic or metaphorical sense. The sentence is correct without it, but is fine grammatically with it as well.

Passing the birch, you look back and see a boy oin the tree!

Even at the very top of the tree, it makes more grammatical sense to say that the boy is "in" it.

Blond, thin, aged six or seven at best, clad in a red T-shirt, beige shirts, and small trainers, he sits in the tree fork at the height of the second floor, ten meters above the ground, and purportedlyappears to looks into his smartphone.

"Purportedly" would be used more in a situation where you are not able to observe the boy your self. (Example: A friend tells you, but you are unable to verify.) It is also a word that is fairly uncommon in everyday, casual conversation.

After several swayings of the birch, the rucksack is on thehis back, and the boy is on the way down, deftly crawling from one tree tier to another.

The boy's possession of his back should be specified in English.

Finally, he completes the last jump, and leaves the grounyard.

If you mean to say that he leaves an area, such as a yard, to walk on asphalt, the term "ground" would not be accurate. It makes it seem as though he is flying. "Yard" or "lawn" might be a better word to use in this context.
The comma makes the two separate ideas flow better together.

Suddenly, you notice an unusual movement in a birch standing byon the side of the road.

Passing the birch, you look back and see a boy oin the tree!

Blond, thin, aged six or seven at best, clad in a red T-shirt, beige shiorts, and small trainers, he sits in the tree fork at thebout two stories height of the second floor, ten meters above the ground, and purportedly looks intoat his smartphone.

After several swayings of the birch, the rucksack is on thehis back, and the boy is on the way down, deftly crawling from one tree tierbranch to another.

Finally, he completmakes the last jump and leavesands on the ground.

Soon, he turns into the arch of thea house and disappears.

"The house" would be a house that we already know about. Since this is a new house that we have no knowledge of, we say "a house."