For your first question, a requirement for a solar system is that there must be at least one star contained within it. Without a star or some other intense gravity field holding the planets in orbit, the planets would drift away. It is possible for any class of star- from dwarf to supergiant- to hold planets in orbit and therefore have a solar system.
Current technology cannot identify magnetospheres around exoplanets due to our limited processes. However it is reasonable to assume that a planet in the habitable zone of a star may possess a magnetosphere.
A planet's magnetosphere is dependent not on a star, but on the rotation of that planet's core. Earth's core rotates within the mantle, creating our supposedly unique magnetosphere. The only real interaction between a star and a planet's magnetosphere is the magnetosphere deflecting solar winds from the star.