Why do planets tend to rotate in the same direction although they have formed from tumbling asteroids? The axial tilts of asteroids seem to vary randomly (let me know if this premise is wrong), while the planets have a strong tendency to rotate in the same way. If planets were formed by colliding asteroids, shouldn't the sum of random tilts result in random planetary rotation? Of course other factors are important, like angle and velocity of impacts, YORP-effect, centrifugal breakup and whatnot, but how could that taken together have any systematic effect on rotation?

Ceres does behave with 4° tilt, but the other of the first discovered asteroids have tilts like 84°, 50°, 42°. Dust particles (and gas molecules if applicable) surely rotate randomly. The Solar nebula had a net spin which gravity and friction has manifested in the orbits of the planets. But shouldn't the netting of rotation be individual for each planet, with uncorrelated tilts, as orbital orientation is for each star?
You are right that the tilt of the asteroids are distributed in very random way, and that the rotation of the Solar nebula is a minor contributor to that tilt, and only skews it a little.

However, you are not right that randomness simply adds up. The randomness does in fact cancel out more and more when you combine a large amount of asteroids, until the rotation of the nebula becomes the dominant factor. This is related to the Law of large numbers.

For instance, throw a dice. The outcome is random. Throw 10 dices, calculate their sum, and divide by 10. Not so far from the average any more? You can do the same thing with thousands of dices, or millions of asteroids. When the number of asteroids that form an object is really high, the tilt is not going to be far from the average value, determined by the nebula's rotation.

The same argument goes for inclination, and the fact that even though the planets' orbits are elliptical, they are not as far from circular that a random orbit would be.