3 months
It takes 28 days for the moon to travel round the earth.
About a second and a quarter.
About 28 days.
About a month
A complete moon cycle, known as "lunation" is about 29.5 days. It ends up back where it started, which I suppose is one definition of "going all the way around the earth". So the answer is one quarter of 29.5, which is one week and 9 hours.
as long as the earth's been round!
The moon rotates round the Earth.
When the moon is in the same direction of the sun, it is called a new moon. A week after the new moon, when the moon has completed about a quarter of it's turn around Earth, this is called the first quarter and about three weeks after the new moon and a week after the full moon, it is called the last quarter. If you visualise the moon turning around the Earth and imagine that the direction of which the sun is shining is called the "front' of the Earth, the the quarter moon would be at the 'sides' of the Earth and the full moon is at the 'back'.
The moon goes around the earth every 27 days and 8 hours. (approximately)
They form a right-angled triangle with the Moon at the right-angle. The Moon is approximately in the direction that the Earth is headed in, in its orbit round the Sun.
No they are not. The moon is round so a half moon would be half of the moon showing while a quarter moon would 1 out of 4 parts of the moon.
The moon does not orbit the sun independently; it orbits the earth. So the moon orbits the sun in about 365 and a quarter days, just like earth.