Is the Moon inside the Earth's magnetosphere?

Yes and no. The magnetosphere is bullet-shaped due to the action of the suns radiation on our magnetic field and as such is quite close, relatively speaking, to the earth on the side facing the sun and quite a bit further away on the far side. The moon will pass through the magnetosphere when it's orbit brings it behind both the sun and earth and pass out of it when it moves more to the sides. Most of the time, the moon is outside of our magnetosphere.