It is any natural body past the orbit of Neptune. (e.g. Pluto, Eris, Sedna)
20000 Varuna is a transneptunian object, meaning it is very far away, and not much is known about it. However, any object of its size is likely to have at least some magnetic field.
Pluto is a dwarf planet. The Pluto/Charon system is the largest known Kuiper Belt Object, or Transneptunian Object. Pluto itself is the second largest known Kuiper Belt object, just behind Eris.
The Kyper Belt is the region far outside the orbit of Neptune where objects, also known as transneptunian objects, orbit the sun. These objects can differ from dwarf planets, like Pluto, to asteroids or other debris. Their origin is believed to be that they are leftovers from when the solar system was very young and were thrown out there by the gravity of the current planets.
First and foremost is the sun. About it are arrayed eight planetary objects, their moons, numerous dwarf planets, an asteroid belt, a Kuiper Belt (largely comprised of icy cometary transneptunian objects), and beyond that an Oort Cloud, or shell of icy cometary debris. Short period comets hail from the Kuiper Belt, while long period comets may come in from any trajectory and originate in the vasty deep--very far from our sun.
Mostly methane ice and rock
How far asteroids can go would very much be a matter of definition. Because of their relatively small size and lack of intrinsic luminosity they would be extremely difficult to locate outside the solar system but there is no reason they could not be in interstellar or even intergalactic space. However, whether they would still be considered and categorized as asteroids might be another matter; they might be referred to as rogue planetoids or rogue planetesimals outside the gravitational influence of the Sun (beyond about 50,000 astronomical units distance). Within the solar system they can be everywhere - the bulk of them lie between the orbits of Mars and Jupiter; beyond Jupiter out to the orbit of Neptune they are known as Centaurs, past Neptune they are referred to as Transneptunian. Fewer are in the inner solar system. and may interact gravitationally (and evidence resonance) with the orbits of Venus or Earth for example; some are in very long-period orbits with the Sun - and just to confuse matters further, recent studies indicate based on composition that the properties of comets may blur somewhat with what we currently think of as asteroids.