Any mass attracts every other mass in the universe proportional to the product of the masses and divided by the square of the distance bewteen them. Both are being pulled by the earth due to the gravitational force. The moon stays in orbit due to it having the appropriate tangential velocity that keeps it from coming closer to the earth's surface. The moon, however, is still accelerating at the rate any object would have at that distance from the earth. Satellites in geosynchronous orbit match the period of the earth in order to look at the same place all the time.
True weightlessness occurs when there is an absence of gravitational force on an object.
This does not occur. Apparent weightlessness occurs when objects are together in freefall and even though they may have a large force on them, they wouldn't register on a scale that was in freefall with them. Astronauts appear this way in the shuttle. Describe apparent weight in an elevator accelerating up, down, and not accelerating. An elevator accelerating someone up must not only apply a force to provide the acceleration but also oppose the gravitational force on the person.