Why do all the planets of the solar system orbit in roughly the same 2D plane?
Nic and Approximist's answers hit the main points, but it's worth adding an additional word on the reason the orbits lie roughly in the same plane: Conservation of angular momentum.
The Solar System began as a large cloud of stuff, many times larger than its current size. It had some very slight initial angular momentum -- that is, it was, on average, rotating about a certain axis. (Why? Maybe just randomly! All of the constituents were flying around, and if you add up those random motions, there'll generically be some nonzero angular momentum.) Because angular momentum is conserved, as the cloud collapsed the rotation rate sped up (the usual example being the figure skater who pulls in her arms as she spins, and speeds up accordingly).
Further collapse in the direction perpendicular to the plane of rotation doesn't change the angular momentum, but collapse in the other directions would change it. So the collapse turns the initial cloud, whatever its shape, into a pancake. The planets formed out of that pancake.
By the way, you can see the signs of that initial angular momentum in other things too: not only are all of the planets orbiting in roughly the same plane, but so are most of their moons, and most of the planets' rotations about their axes as well.
More or less yes. The planets mostly orbit in the same plane but with small deviations compared to the size of the system. The largest relative tilt is around 4 to 6 degrees. This 'flatness' is due to orbital mechanics, where the solar system started spherical it now has 'decayed' into a flat disc. This disc is essentially a stable low energy configuration and therefore the system has become more and more disc like over time.
The Voyager 1 probe is 'leaving' the solarsystem as we speak but there is no real firm boundary and so it's hard to give a definitive answer.
The orbital planes of different planets has small inclinations to the ecliptic plane. The corresponding wikipedia diagram should give a better view than the 2D images you've seen.
Both Voyager 1 and 2 are beyond the Solar System