Yes, the Earth would have phases just like the moon.
The interesting thing about this is that the moon is tidally locked to the Earth. What this means is that the moon rotates only once on its axis in the time it takes to orbit the Earth. One "day" on the moon is about 28 days on Earth. A consequence of this is that if you stay in one location on the moon, the Earth is always in the same part of the sky. If you landed on the "dark" side of the moon you would never see the Earth at all. Wherever you are on the moon, the sun would rise every fourteen days or so and set about fourteen days later.

Ah, an excellent question. Your students aren't old enough to remember the first photos from the Apollo capsules as they orbited the moon and captured a picture of an "earth rise", but clearly they are thinking about the same perspective.

Yes, the earth has phases when viewed from the moon. And yes, they are opposite from those seen on earth. You can prove it with two balls and a flashlight in a darkened room. Put the flashlight on a table, and aim it at another table with the two balls on it. Remember that the moon orbits around the earth about every 28 days. Put the moon farther away from the "sun" flashlight than the earth, and look at it from directly over the earth.
You should see the moon mostly illuminated from this point of view (like a full moon). Now, look at the "earth" from over the moon. You should see only part of the earth lit up, since it is closer to the sun. Now you're seeing a "new earth" with the same perspective as when you see a new moon.

Try reversing the position of the balls and changing your viewpoint accordingly.

Now, what happens when the earth is directly in front of the moon, blocking the light from the sun? And when the moon blocks the sun's rays from reaching the earth? What do we call these special cases?