Yes, it is commonly thought the moons originated as asteroids, pulled from the belt by gravity.
Scientists think that the asteroid belt is a group of planetesimals (solid objects) that are caught between the gravity of Jupiter and the Sun.
gravity
A planet
The sun
Jupiter
the gravity from Jupiter is so strong that it pulls most of the comets and asteroids to it instead of the inner planets
The asteroid belt, is a "collection" of asteroids circling the Sun in between the orbits of Mars and Jupiter. A collection cannot have an atmosphere. A single asteroid does not have enough mass to "hold" on to an atmosphere.
Jupiter has an effect on the orbits of the asteroids, but I wouldn't really say that's what holds them in the area beyond Mars.
Some asteroids are considered associated and are called "groups." Most asteroids are not part of groups but are in a "belt" positioned between Mars and Jupiter. Certain zones of gravitional equilibrium can capture asteroids. These zones are known as LaGrange Points. Jupiter, having the most powerful planetary gravity, has several groups at these points. Two of them, because the individual asteroids in them are named after heroes in the Trojan War, are known as "Trojan asteroids." One group is the "Trojan" group and another is the "Greek" group, even though there's an important Trojan in the Greeks and an important Greek with the Trojans.
The asteroids stay together because of gravity, because of orbital stability around the sun, and because of the graviometric influence of Jupiter, the other planets, and each other.
No, the gravity between Jupiter and its moons acts towards the centre of Jupiter.
The small rocky bodies between Mars and Jupiter are called asteroids. These objects are small and are made of rock or iron. Jupiter's gravity is stopping these rocks from forming a planet but even if they did form a planet, the resulting planet would only be half the size of the moon.