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Speed of Earth:
The circumference of the Earth at the equator is 25,000 miles. The Earth rotates in about 24 hours. Therefore, if you were to hang above the surface of the Earth at the equator without moving, you would see 25,000 miles pass by in 24 hours, at a speed of 25000/24 or just over 1000 miles per hour.
The moon doesn't rotate around the equator — it's inclined by about 20 to 30 degrees. So, no, it won't necessarily appear at the same height in the sky; in fact, it stays within about 5 degrees of the sun's path (the ecliptic).