Answer 1:
In theory, you can use Newton's law of Gravitation
to predict the required conditions for a planet to
orbit at the speed of light. A rough approximation
of this gives us that the velocity is given by
v= G M/r, where r is the
orbital radius and M is the
mass of the body that the planet is orbiting
around.
If we want this velocity to be the speed
of light, we can set v = c and solve for
the required radius, which gives us
r = G M/c^{2}. To get
an idea for how small this radius is, for the mass
of our sun, the planet would have to be orbiting
just around 1.5 km away from the center of the
sun. So, this scenario is not possible of an
orbiting planet moving at the speed of light, is
not possible. However, an example of something
similar happening is that light orbits around a
black hole at the speed of light.
Thanks!

Answer 2:
No.
Here's a good source  Scientific American  for
news about neutrinos that travel faster than light
in a special container. As far as I can tell, the
results are still correct.
Particles Found to Travel Faster Than Speed of
Light.
Sep 22, 2011  "An Italian experiment has unveiled
evidence that fundamental particles known as
neutrinos can travel faster than light. Other
researchers are ..."

Answer 3:
No, the gravity of a planet is not strong
enough to accelerate past the speed of light.
The only class of object whose gravity is strong
enough to accelerate an object past than the speed
of light are black holes, and that only
happens inside of the event horizon  which
means that whatever does go faster than light
inside of a black hole will not come back out.
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