On what planar regions are the Earth, Moon.. for that matter, all astronomical bodies, placed in the Universe? When we are on Earth, we look UP in the Sky to see the Moon. How do we have to look at Earth from Moon.. Is it the way ? If so, how are these bodies actually placed in the space?
Are all astronomical objects lined up on the same planar level? When we look at stars, are they actually above the Solar System planar level or below ?!
Because, in general, if we have to look up to see something on a building, then from the building we have to look down to see the same object. 
 A: The concepts of "up" and "down" do not make sense in space; they only have meaning in the dense gravitational field of a large mass like a planet or a star. The universe is three-dimensional: galaxies are distributed in three dimensions. Many, but not all, galaxies have an internal structure like a swirling disk of stars, relatively flat in one dimension, but as you zoom in, the stars within a galaxy are more or less distributed three-dimensionally. It is only the planetary systems around stars that are primarily two-dimensional, because of their origin in a spinning cloud of matter.
Within a "gravity well" you have up and down. From the surface of the Earth, we look "up" at everything in the sky: Sun, Moon, planets, stars, other galaxies. The only thing "down" is the ground beneath our feet. On the surface of the Moon, or Mars, everything in the sky is also "up," and only the ground beneath our feet is down.  The local gravity field trumps everything else, as far as our perceptions are concerned.
A: The planets in our solar system rotate around the sun on a plane called "ecliptic". It isn't an exact plane but the orbits are rotated slightly against each other. Since moons rotate not far from their planet, you can count them, too, as inside the ecliptic. Everything else is generally on a different plane. The universe is in 3D.
A: Coplanar orbits are very common in astronomy. From the small scale (moon orbits) to the large scale (galaxy discs), and this is due to the conservation of angular momentum. If you have a cloud that is being exerted upon by force (gravity), the conservation of angular momentum will cause an initial spin to be amplified. As the cloud spin speeds up, it flattens out into a disc, which is the maximal balance between gravitational collapse and centrifugal force.
In the sense of our solar system, the planets were formed through the aggregation of particles spinning in a big disc, these particles coagulating together to form the planets we know. That explains why they all follow essentially the same orbit plane (with the exception of the planetoid, Pluto).
When you look at the bigger scale, say a galaxy, you'll see this coplanar orbit mechanism persist through disc-shaped galaxies, such as the Milky Way, or Andromeda... even the spinning accretion disc of a black hole.
