1.The concept of a month comes from a root of mo. Month and Moon both have Mo. The ancient people came up with months because the Moons cycle takes about 1 month.
2. You can see the Moon during the day sometimes. You can see the Moon during the morning sometimes because the moon just follows the sun so when the moon is following the sun so if the moon is above the horizon then you can see it.
3. A full moon is the only phase of the moon that can only be seen at night. It can only be seen at night because a full moon is the only phase that at night when the moon is directly across from the sun. So basically right when the sun sets the moon rises but when the sun rises the moon sets.