What was the primary role of dust in the formation of the solar system?
In interstellar space, there is gas and dust. Gas, and primarily hydrogen, is the fuel for stars - the gas condenses together and as it does so begins to heat up until the heat is high enough to jumpstart atomic fusion reactions (where 2 hydrogen atoms collide with such an impact that they bond and form helium. And then hydrogen and helium bang and collide into other atoms and as they stick together form more complex and denser atoms, such as iron, gold, and uranium. In fact, all the elements on the periodic chart, up to uranium, are made in stars.)
Once a star's fuel is depleted to where the heat expansion from the fusion reactions overcomes the gravity attraction holding the whole thing together, it can (if it's big enough) explode into a supernova - spreading dust, or all of the elements it has made, into the surrounding area.
The solar system is built with the dust of a prior supernova - the dust that the previous star (or stars) blew out condensed into the Sun, the planets, the moons... everything.