when we see the moon in the day, does this mean the people that are on the opposite side of the world cant?

That's correct. At any given time, about 1/2 the Earth can see the moon. If you see it, then someone on the direct opposite side of the world cannot.

So, if you see the moon in the day - so does the rest of the world. The moon slowly orbits the Earth (once in 29 days), and while it's in that position, the Earth turns below it - so everyone sees the moon in (about) the same position as you do. When it's a full moon - everyone sees a full moon; but for those experiencing noon during the full moon, it's below the horizon - they have to wait until sunset to see the moon rise over the horizon.

The Moon is at the same phase everywhere in the world, but only half the world can see the Moon at any one time. There is nothing unusual about the Moon in the daytime sky: it's there half the time if you bother to look for it. When it is visible in your daytime (say in the UK), it will be below the horizon on the other side of the Earth. 12 hours later, the Earth will have rotated 180°, and the Moon will be in _their_ daytime sky, and below the horizon where you are. You just need to realize that the Moon is very far away and moving very slowly around the Earth, while the Earth is rotating on its axis quite rapidly. Now you see it, now you don't.

That is correct, since the moon can't be in 2 places at once.

Think about it - there are nights when you don't see the moon, that's because its visible over another part of the world.