Nrthugu

I know the interstellar space is not empty. It has (among micrometeorites and gas clouds) interstellar dust. And the effect of the dust is more noticeable if the speed of the starship increases (we are talking here about speeds not faster than light). My question is: At which speed could an interstellar ship use the interstellar dust to maneuver through it with flaps and ailerons? (like a plane does with the air).

It is not only how fast the airfoil has to travel, but also how large the airfoil must be to even have fluid-dynamics-like behavior. Because of the low density of particles in the interstellar medium, a normal-sized wing will not act as an airfoil. It will be more like bouncing tennis balls off the wing every so often.

The Knudsen number quantifies when fluid dynamics takes over: Kn = mean_free_path / airfoil_length_scale. Knudsen numbers greater than about 10 are in the regime of ballistic collisions, rather than fluid flow. The mean free path in the interstellar medium is huge (about 70 astronomical units). So to get fluid dynamics, you'd need an airfoil of size more than the Sun–Jupiter distance.

What is the difference between the fluid dynamics (low Knudsen number) and the ballistic regime (high Knudsen number)? When you are driving down the road and air flows over your windshield, this is because some air molecules collide with the windshield and then collide with air molecules further in front you, transmitting force forward. In a sense, air molecules several meters in front of you know about your car and begin moving away before the car actually reaches them. High Knudsen number, is, again, like driving through a field of floating tennis balls. They bounce off your windshield, but don't collide with or transfer their momentum to any of the other tennis balls (assuming the tennis balls are widely spaced). You can change your momentum by deflecting them at different angles, but it won't work anything like the flaps and ailerons of a plane.

At relativistic speeds, you might be able to steer by deflecting molecules of the interstellar medium. Even at these speeds, the drag force (and your ability to apply steering forces) will be very low for a normal-sized wing. There's a calculation on reddit for for drag in the interstellar medium at 0.9c. See Russell Borogove's answer for practical issues with utilizing this drag.

The density of the interstellar gas and dust medium varies widely, but I estimate that for fairly dense regions (1 million hydrogen atoms per cc), you get about 1 Newton of lift from a meter-square area of wing at about 10% of the speed of light, so the concept isn't totally inconceivable.

The trouble here is that friction with the interstellar medium is a real problem. You're hitting matter at 10% of the speed of light, which is going to produce a lot of heat. It's generally assumed that some sort of non-physical shielding is going to be necessary for high-sublight speeds -- perhaps a big laser to ionize the gas followed by a big magnetic field to push the ionized gas out of the way (or to collect it for a Bussard ramjet).