Why does Pluto's orbit cross Neptune's orbit? Why does Pluto's orbit cross Neptune's orbit? Is this a fault in Newton's law of gravitation?
 A: Pluto's average distance from the Sun is larger than Neptune's but Pluto's orbit has a higher eccentricity – the elliptic orbit is more squeezed, less uniformly circular, and such ellipses simply do intersect each other.

The elliptical orbits with properties first identified by Kepler's laws do follow from Newton's laws of gravity. I stress that the orbits are general ellipses, not necessarily circles.
A: The simple answer to Why does the Pluto's orbit crosses the Neptune's orbit is to just say that's the way it is. For any object orbiting in a central inverse square law field, like the gravitational field of the Sun, the stable orbits are ellipses with the Sun at one focus. The ellipses can be almost circular like the Earth's orbit or wildly eccentric like the orbit of Halley's comet: both are stable orbits in Newtonian gravity. So Pluto could have any elliptical orbit and it would be a perfectly good solution to Newton's laws.
However there is more to it than this. Orbiting objects don't just feel the gravity of the Sun, they also feel the gravitational fields of all the planets and indeed all other massive bodies in the Solar System. In general if you put an object into some randomly chosen orbit that crosses other orbits it will eventually come close enough to one of the massive bodies to interact strongly and be ejected from the Solar System. If you have a play with a gravity simulator (My Solar System is one I like) and add a light body to a system with several planets you'll probably be surprised how unstable most orbits are.
The orbits of the planets we see today are stable because they're the ones that lasted 4.5 billion years, all other orbiting bodies having been ejected. Even so the orbits are less stable than you might think. For example the ellipticity of the Earth's orbit changes continually in a cycle of 96,600 years due to interactions with other planets (mostly Jupiter).
Anyhow, back to Pluto. We can't say why Pluto's orbit has exactly the shape it does. However Pluto's orbit is not randomly chosen - it has a resonance with the orbit of Neptune: for every two orbits that Pluto makes around the Sun, Neptune makes three. Without this resonance it's likely that Pluto would eventually come close enough to Neptune for it's orbit to be wildly perturbed and it could even be ejected from the Solar System. The point is that we can't say why Pluto crosses the orbit of Neptune, but Newton's laws tell us that if it does have a crossing orbit that orbit must be a special one or it wouldn't be stable.
A: Short, no-math answer:
Objects in space can have any orbit they want to have. How long they stay in that orbit depends on what else is out there. Over the past 5+ billion years the various interactions have sorted themselves out and we see what's left.
There's no reason a planet could not be in a retrograde orbit, a perpendicular orbit, or a figure-eight loop between Jupiter and the Sun. I do not expect those orbits would be stable or survivable, so you won't see them during the 0.00001% of the solar system's existence we've been looking at it.
A stable figure-8 perpendicular orbit between a star and it's first gas giant would be really cool though.
A: All orbits are ellipses.
Ellipses can be perfect circles, but are always "stretched" to some extent.  The Sun (in this case) will be at one focus of the ellipse.

Thus a very stretched (more elliptical) orbit of one body (e.g. Pluto) can be sometimes be closer to the Sun than a less stretched (more circular), but closer to the sun orbit of another body (e.g. Neptune).
