|MadSci Network: Astronomy|
You ask some very good questions: the answers could easily take up a small book!
To tackle your first observation: yes, a reasonably good representation of the Solar System can be made on a flat sheet. Most of the planets in the Solar System orbit very close to a plane that is defined by the Sun's equator, so their positions can be roughly approximated with a view taken perpendicular to that plane. The distances between the planets is almost irrelevant; it's the similarity in orbital plane that allows this.
However, the planets do not all lie exactly on that plane, the most notable being Pluto, which has an orbit inclined from that plane by 12 degrees, and the recently-discovered body currently designated 2003 UB(313), which is larger than Pluto, has an orbit skewed by almost 45 degrees. Looking at a two-dimensional representation will give a somewhat inaccurate idea of the locations of most of the planets and wildly inaccurate picture of the positions of these two bodies.
The same sort of situation exists with our galaxy. Most of the visible material in the galaxy is concentrated very close to its plane of rotation, giving the appearance of a flat disc, but there are bodies orbiting the core of the galaxy which are pretty evenly distributed in a spherical volume all the way around it. Many of those bodies follow orbits around the galactic center that are inclined as much as seventy or eighty degrees from the rest of the visible galaxy. No single two-dimensional view can accurately locate all the objects moving around the center of the galaxy.
If we expand our view to include the entire universe, we find a roughly spherical bubble about twenty-four billion light years across, give or take a few billion light years. The many galaxies and other bodies inside the universe are distributed fairly evenly throughout that sphere, with minor variations in density from place to place. Because the matter in the universe---galaxies, stars, and whatnot---is so uniformly distributed in three-dimensional space, once again a single two-dimensional view cannot give an accurate idea of a body's location.
Now, about up and down ... "up" is defined by our experience as the direction opposite the pull of Earth's gravity. To generalize, it's defined as the direction away from the center of mass of a planetary or stellar body. "Down" is defined as pointing towards the center of mass of that same body. This works pretty well on a planet's surface, but it doesn't work at all if you bring more than one celestial body into the picture. The direction "down" to someone standing on the Moon is referencing a completely different point in space---the center of mass of the Moon---than the direction "down" to someone standing on the Earth. Since there are multiple points in space which can be used as a reference for "up" and "down"---one for every celestial body in the universe, in fact---and none of those references is useful away from that particular body, the terms "up" and "down" are generally meaningless when looking out into space.
What is useful is to choose an agreed-upon reference point in space and then define directions relative to that point. An earlier answer I posted some time ago discusses how such references can be used to describe any location in space. That answer has several links to websites which discuss reference systems.
If you'd like to see a little more about the structure and shape of the universe I would recommend The Illustrated Brief History of Time and The Universe in a Nutshell, both by Stephen Hawking. Dr. Hawking presents a very good, non-mathematical treatment touching upon many of your thoughts concerning space and the universe.
I hope this has been helpful!
Try the links in the MadSci Library for more information on Astronomy.