Answer 1:
To understand things moving at the speed of light,
we refer to Einstein's Theory of Special
Relativity. Einstein invented the idea of
spacetime: thinking about time as just another
dimension. What makes it hard to imagine is the
fact that we can't "see" it (time).
The
speed of light is a constant and it is measured in
time always (299,792,458meters/second or 186,000
miles/second). Light changes its speed when it
moves from one medium (say air) to another (say
glass), but it is still measured in distance units
per time. The reason why light changes its speed
is that it consists of electromagnetic waves that
interact with the atoms of the medium through
which it travels. This phenomenon is called
refraction, and slows down the speed of
light.
What we usually call the speed of
light is really the speed of light in a vacuum
(the absence of matter). In space, the density of
matter between the stars is sufficiently low that
the actual speed of light through most of
interstellar space is essentially the speed it
would have through a vacuum, so we can ignore the
difference. You can also think about the earths
atmosphere and the space as different mediums, but
the light traveling through them, still will be
measured in time, and will be a
constant.
If you are really interested in
this topic, try to read on the next address,where
you will find interesting things about the speed
of
light:http://csep10.phys.utk.edu/guidry/violence/lightspeed.html

Answer 2:
First, the Lorentz transformation and time
dilation is a result of coordinate shifts in
relativity. At some level, it does not make sense
to talk about photons "experiencing"
time.
Nonetheless, you are right: if you
are not traveling at C (speed of light), anything
that is traveling at C could be said to have
infinite time dilation, i.e. for a given period if
time that passes for you, no time passes for the
object traveling at C. In this way, photons are
timeless entities  time does not exist for them
because they always travel at C.
Now, at
some point somebody is going to ask what happens
if a photon starts moving through a medium other
than vacuum (and thus has an index of refraction
other than 1). In this case, the photon is
interacting with the substance it is passing
through, even being absorbed and continually
reemitted. The medium certainly does experience
time.
Now, your second
question:
What your student quoted is one
of the principles of general relativity. The idea
here is that spacetime is a curved,
fourdimensional hyperspace, with time being one
of the dimensions, but the geometry is Riemannian,
not Euclidean. Namely, the distance,called the
"interval" between two points is the square root
of the sum of the squares of the distance in x,y,
and z dimensions, MINUS the square of the distance
in the time dimension (and the distancetime
conversion factor is c). Under this
calculation,anything that is traveling at the
speed of light will actually travel zero distance,
since the sum of squares of the distance traveled
in the space dimensions will equal the square of
the time passed.Anything moving less than the
speed of light will have an imaginary interval,
because square of the time passed will be greater
than the space distance traveled, and so the
interval will be the square root of a negative
number.
I think that what your student
found was a reference to the imaginary sums of
distances and of times. I unfortunately cannot
explain any further how the interval works in a
physical sense. A complete understanding of
Einstein general relativity requires an
understanding of mathematics that you normally
don't get until you are in graduate school, and my
degree is in paleontology, not physics. Photons of
course are a quantum phenomenon, and I think that
I can safely say that whoever figures out how
general relativity fits together with quantum
mechanics is going to get the Nobel Prize.
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