Assuming that the observer is on the moon for a full lunar cycle, would the appearance of the planet wax and wane, would the moon's shadow occlude the planet during an eclipse while on the moon?
The planet, as seen from the moon, would undergo a cycle of visibility, going from full "planet" to new "planet" and back, depending on the mutual position with respect to the star, like in the following examples taken from the Moon:
About casting its shadow on the planet, it can happen, too, if the shadow cone intersects the planet body.
Stellarium is an awesome open-source project if you want to answer those kind of questions.
You can specifiy the location and time you'd like to observe the sky. The location isn't always on Earth, you can also pick the Moon, Jupiter, Mars, Saturn or any of their moons!
Solar Eclipse on Earth, seen from the Moon
Moon's shadow on Earth during an eclipse should be visible with the naked eye. Earth as seen from the Moon is 4 times larger than the Moon as seen from Earth.
Lunar Eclipse on Earth, seen from the Moon
- Note that the Earth doesn't move at all in the Lunar sky, due to "tidal locking".
- "New Earth" always happens during the lunar day (i.e. "Full Moon").
- "Full Earth" always happens during the lunar night (i.e. "New Moon").
- One lunar day is what we call a month.
- During a lunar day, Earth rotates on itself approximately 29 times (while staying at the same position in the sky), so every part of Earth will be visible multiple times.
I understand this has been pretty thoroughly answered as a decisive "Yes", but I wanted to offer another literal point of view.
As you can see in the image below, the sunlight always comes from the same direction for both the sun and the moon. You can see that as the moon orbits the earth (a lunar day), different lighting conditions exist along the way. They are in fact opposite of the current phases of the moon, which I think is just neato.
In the first quarter, standing exactly in the middle, you can see the opposite phase of the earth. In this case, what looks like the third quarter. As the moon orbits counter clockwise to the Waxing Gibbous, the earth's new phase is directly opposite in the graphic, meaning it is now Waning Crescent from your point of view. Follow one more phase to when it's a full moon (Lunar "noon"), you can see that the earth is now covered in shadow. A "New Earth" if you will. You can continue along the path in the chart below to see how the moon orbits and how the faces change. In the moon-sky, the light and shadow would move "right to left" the same way our moon does in our earth-sky.
Now, we also note that the earth is spinning once every 24 hours. But this has no bearing on how we see it as far as the direction of light goes. Normally we take this for granted, that the moon is tidally locked with the Earth. So as the phases of the Earth progress, you may notice that different parts of the Earth, different features like countries and oceans, are visible to you. It may even appear to "rock" back and forth as the seasons change (the phases still appear normally). But how the Earth spins has no bearing on which parts of it are lit.
A "full planet" occurs when the moon is between the planet and the star. A "new planet" occurs when the planet is between the moon and the star. So as long as which one is closer to the star is changing, the planet will wax and wane. For the planet to not wax and wane, you'd have to have a really weird situation, such as that the moon's plane of rotation is perpendicular to the line between the planet and star.
Note that a perfect full planet would mean that the moon is exactly between the planet and the star, which would mean that the moon's shadow is being cast on the planet, so the planet would see a solar eclipse, and the moon would see a planet eclipse. In a perfect new planet, the moon would see a solar eclipse and the planet would see a lunar eclipse. Whether those would be total eclipses depends on their size and distances. What we call a "full moon" is not a perfect full moon, as the moon is usually not on exactly the same plane as the earth. When it is on the same plane, and the moon is opposite the sun, we get a lunar eclipse rather than a full moon.
Are you talking only about the Earth and the Moon? If you are talking about any moon around any planet, then the view would also depend on the plain of rotation of the moon around the planet.
If the moon orbitted on a plane perpendicular to the planet's orbit around it's star - imagine the star to the left, the planet in the centre and the moon (appears to be) moving up and down - then it would only see the normal "wax" and "wane" when it crosses the planets orbit, at other times (especially when position above or below) the planet would just be half/half all the time.