> The sphere of influence in which the gravitational attraction of the asteroid dominates over that of the sun has a radius of about 6km though.
It depends on what you consider "dominates" to mean. For example, the corresponding value for the Earth and the Sun is 924,000 km, and the Moon is only 400,000 km from the Earth, but the Moon's orbit is still always concave towards the Sun.
If Earth wouldn't dominate the space in which the Moon moves, the Moon wouldn't be in orbit around Earth at all and the Sun couldn't perturbe this orbit (as you even stated yourself)?
> If Earth wouldn't dominate the space in which the Moon moves
My point is that "dominate" here could mean different things, and the Earth does not dominate all of them.
Specifically, the fact that the Moon's orbit is always concave to the Sun means that the "acceleration due to gravity" of the Sun on the Moon is larger than the "acceleration due to gravity" of the Earth on the Moon. In other words, the net "acceleration due to gravity" of the Moon is always towards the Sun, not towards the Earth. So the Earth does not dominate in this way.
The "sphere of influence" you mention is based on tidal forces: the tidal effect of the Earth on the Moon is larger than the tidal effect of the Sun on the Moon. So the Earth does dominate in this way.
> the Moon wouldn't be in orbit around Earth at all and the Sun couldn't perturbe this orbit (as you even stated yourself)?
No, that's not what I said. Whether you think of the Moon as orbiting the Earth or the Sun depends on how you define "orbit" and what you are trying to do. If you define "orbit" according to which body the Moon is accelerating towards, on net, then the Moon is orbiting the Sun, not the Earth (see above); in this sense, the Moon and the Earth are in two closely matched orbits about the Sun.
And my point about the Hayabusa spacecraft and the asteroid is that, if you calculate the "acceleration due to gravity", you find that it's similar to the Moon's--the spacecraft's net acceleration is towards the Sun, not towards the asteroid.
It depends on what you consider "dominates" to mean. For example, the corresponding value for the Earth and the Sun is 924,000 km, and the Moon is only 400,000 km from the Earth, but the Moon's orbit is still always concave towards the Sun.