To us on the Earth, the moon travels across the sky, and goes through
a 30 day cycle of invisible to fully visible.
 
How would the Earth appear to astronauts on the Moon?   
A) It would follow a similar 30 day cycle as it travels across the lunar sky. 
B) The Earth would remain stationary in the lunar sky and always show
the same position; no cycle, no movement, no rotation of any kind.
C) The Earth would remain stationary in the lunar sky while rotating
and going through a 30 day cycle.
D) The Earth would follow a 24-hour cycle from invisible to fully
visible as it travels across the lunar sky in a 30 day cycle.
E) None of the above.
 
How exactly did Neil Armstrong describe the Earth to the world in 1969? 
  
When the earth is between the moon and sun, would the earth be too
dark to see, or would you see a ring of light from the atmosphere?

The earth, as viewed from the moon would also go thru phases.  This
link shows earthrise as viewed from the moon.

http://antwrp.gsfc.nasa.gov/apod/ap051224.html

It was taken by the astronauts onboard Apollo 8.

I didn't see your second question when I gave my first comment.  When
the earth is between the moon and the sun we would have a lunar
eclipse as viewed from the earth and a solar eclipse as viewed from
the moon.

During a solar eclipse on the earth, the moon moves between the earth
and sun.  It is about the same apparent size in the sky as the sun, so
it is just big enough to block the sun during a total eclipse.  At its
maximum, a solar eclipse can last about 7 minutes.  When the moon is
at its apogee (i.e. the farthest point in its orbit from the earth)
its not quite big enough to completely cover the sun and you get an
annular eclipse (the sun covered by the moon with a ring of fire
around the edge).

The earth is considerably bigger than the moon, so it is easily big
enough to block the sun when viewed from the moon.  A solar eclipse on
the moon will last about as long as a lunar eclipse on the earth.  The
maximum duration of a lunar eclipse on earth (or solar eclipse as
viewed from the moon) is about 1 hour, 47 minutes.

http://sunearth.gsfc.nasa.gov/eclipse/LEmono/TLE2004Oct28/TLE2004Oct28.html

The earth would remain rather stationary in the sky from a viewpoint
on the moon since the moon always has one hemisphere towards to earth.
I expect that the earth's atmosphere would allow it to always be
visible from the moon  - perhaps with the exception of when it is very
near the sun.

Just to add to the previous comments.

Since the moon's rotation is locked to it's revolution around the
earth, the earth will appear in approximately the same location in the
sky for an observer in a fixed position on the moon. The position will
move around a little bit, since the moon's rotation axis is slightly
tilted relative to the axis of its orbit and since the orbit is
elliptical, not circular. These are the same reasons why we can see a
bit more than 50% of the surface of the moon from the earth.

In a cycle of 24 hours and 50 minutes you will see the earth rotating
on its axis. The rotation takes 50 minutes longer than a day, since
the moon is revolving in the same direction as the earth's rotation.

We would observe phases of the earth on the same cycle as the lunar
month. When the moon is new, we would see a full earth. When the moon
is full, we would have a new earth.

During a lunar eclipse as viewed from the earth, the moon is well lit
by the light that scatters off the earth's atmosphere. From the moon
we would observe a beautiful ring of light around the earth.

As mentioned in the previous informative comments, the earth would go
through phases and its rotation would show.

If you are interested in seeing the earth as it appears from the moon
and in seeing the rotation and the phases of the earth from the moon,
there are some sites you can go to:

* You can see a current view of the earth from the moon at

http://www.smeter.net/propagation/views/earth-from-moon.php

which also has a comment on its relevance to earth-moon-earth bounce
communications.

* You can also see the rotation of the earth and the phases of the
earth as seen from the moon at another site if you open multiple tabs
and enter different times (3 hour differences work fine for seeing the
rotation - then you can switch from tab to tab to see the rotation).
Changing the day, rather than the time, will show the phases of the
earth, with about a 30 day period; the site even shows the glow from
cities in the dark part of the earth (although I don't know how they
calculated the brightness as seen from the moon).

You can get to this customizable view of the earth from the moon
starting at the site:

http://fourmilab.to/earthview/vplanet.html

and clicking on "view the Earth ... from the Moon." Note: to change
the day or time, I had to change the day and/or time, tab to the next
field and then click on the "update" button.

I was trying to reason about the "path" of the earth in the moon's sky
- there are places on the moon where the earth is always visible and
places where it is sometimes visible, but I ran into an "interesting?"
peripheral issue when I decided to see who devised the multiple choice
part of your question.

The question is from science challenge #26, but what was unexpected
was that another part of the site seems to support the idea that the
moon landing was faked.

http://www.erichufschmid.net/Science_Challenge_26.html

Anyway, the parts of the question that you are interested in are fascinating.

You will see photos labeled as "earthrise," but I believe that these
are not relevant to your question. As best I can tell, such photes
were taken from an orbiter above the moon, not on the moon itself.
Some are taken from Apollo 8, which did not land on the moon. For
Apollo 11, at least one earthrise photo is accompanied by the caption
"Earthrise viewed from lunar orbit prior to landing."

http://www.hq.nasa.gov/office/pao/History/ap11ann/kippsphotos/apollo.html

In addition to what's been said, though Earth will stand approximately
stationary in the Lunar sky, the sky itself, the system of
constellations, will parade behind it in a cycle, each month. This is
just as it does with respect to the Moon as seen from Earth over the
same period of time. (It also happens with respect to the Sun as seen
from either Earth or the Moon over the course of an entire year.)