Pluto

1. Is Pluto a planet?

No, Pluto is no longer considered a planet like the other eight planets in our solar system. The body that decides the classification of objects in the solar system, the International Astronomical Union (IAU), has voted on a new scientific definition of "planet" which states that a planet is a celestial body that:
(a) is in orbit around the Sun
(b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and
(c) has cleared the neighbourhood around its orbit. Pluto does not meet the third criterion, it has not cleared the region of its orbit of other "competing" bodies, so it has been reclassified, along with the asteroid Ceres and the Kuiper Belt Object 2003 UB313, as a "dwarf planet" (which even though it has the word "planet" within it is not considered to be a type of planet).
For more details, see the IAU resolution.

2. Are there any planned missions to Pluto?

Yes. The NASA New Horizons mission is scheduled to launch in January 2006 and reach Pluto in 2015.

3.Is there a "Planet X", a planet beyond the orbit of Pluto?

No, not officially. It turns out that there's a large belt of comets orbiting beyond Pluto, called the Kuiper Belt. One object has already been discovered in this belt that is reliably estimated to be larger than Pluto, but it has not been officially designated a planet. It seems likely many more bodies larger than Pluto will be discovered. It's possible that the object Chiron, currently inside the orbit of Saturn, was one of these objects which escaped. More information on Chiron and the Kuiper Belt is available.